Practice · Fix primes by sum, exponents by divisor count · Sensim Math

Variant 1 easy

Find the largest three-digit natural number that satisfies both conditions below.

(a) Its prime factorization has exactly $3$ different prime factors, and these prime factors add up to $12$ .

(b) It has exactly $12$ divisors.

Your answer

Show solution

1 · Understandwhat's really being asked

A three-digit number has exactly three different prime factors that add up to $12$ , and it has exactly $12$ divisors. Find the largest such number.

Givens

Exactly $3$ different prime factors.
Those primes add up to $12$ .
Exactly $12$ divisors.
The number has three digits.

Unknowns

The largest such three-digit number.

Constraints

If $N = p^a \times q^b \times r^c$ , then $N$ has $(a+1)(b+1)(c+1)$ divisors.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #3 Eliminate Possibilities#7 Identify Subproblems

First pin down which three primes are possible, then use the divisor-count formula to find the exponents, and finally list every candidate and keep the largest three-digit one.

3 · Execute3 carry out the plan

1Find the three primes

#3 Eliminate Possibilities 4.OA.B.4

Three odd primes would add to an odd number, but

12

is even, so one prime is

2

. The other two add to

10

and must be different:

3

and

7

(

5 + 5

repeats a prime).

2 + 3 + 7 = 12

The prime factors must be

2

,

3

,

7

.

2Find the exponents

#7 Identify Subproblems 6.EE.A.1

Write

N = 2^a \times 3^b \times 7^c

with

a

,

b

,

c

at least

1

. Each factor

a+1

,

b+1

,

c+1

is at least

2

, and their product is

12

.

(a+1)(b+1)(c+1) = 12 = 2 \times 2 \times 3

Two exponents are

1

and one exponent is

2

.

3List the candidates

#2 Make a Systematic List 6.EE.A.1

Put the exponent

2

on each prime in turn.

2^2 \times 3 \times 7 = 84, \quad 2 \times 3^2 \times 7 = 126, \quad 2 \times 3 \times 7^2 = 294

The largest three-digit one is

294

.

Answer: 294

4 · Reviewdoes it hold up?

$294 = 2 \times 3 \times 7^2$ : three different primes with $2 + 3 + 7 = 12$ , and $(1+1)(1+1)(2+1) = 12$ divisors. It has three digits.

Another way: To make the number as large as possible, give the square to the largest prime, $7$ , right away: $2 \times 3 \times 49 = 294$ .

Standardsmin grade 6

4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Recognizing which numbers are prime and choosing the three prime factors.
6.EE.A.1 Write and evaluate expressions with whole-number exponents — Writing and evaluating the numbers with exponents and counting divisors from the exponents.

💡Takeaway. The sum picks the primes; the divisor count picks the exponents.

Variant 2 easy

Find the largest three-digit natural number that satisfies both conditions below.

(a) Its prime factorization has exactly $3$ different prime factors, and these prime factors add up to $14$ .

(b) It has exactly $12$ divisors.

Your answer

Show solution

1 · Understandwhat's really being asked

A three-digit number has exactly three different prime factors that add up to $14$ , and it has exactly $12$ divisors. Find the largest such number.

Givens

Exactly $3$ different prime factors.
Those primes add up to $14$ .
Exactly $12$ divisors.
The number has three digits.

Unknowns

The largest such three-digit number.

Constraints

If $N = p^a \times q^b \times r^c$ , then $N$ has $(a+1)(b+1)(c+1)$ divisors.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #3 Eliminate Possibilities#7 Identify Subproblems

First pin down which three primes are possible, then use the divisor-count formula to find the exponents, and finally list every candidate and keep the largest three-digit one.

3 · Execute3 carry out the plan

1Find the three primes

#3 Eliminate Possibilities 4.OA.B.4

Three odd primes would add to an odd number, but

14

is even, so one prime is

2

. The other two add to

12

and must be different:

5

and

7

.

2 + 5 + 7 = 14

The prime factors must be

2

,

5

,

7

.

2Find the exponents

#7 Identify Subproblems 6.EE.A.1

Write

N = 2^a \times 5^b \times 7^c

with

a

,

b

,

c

at least

1

. Each factor

a+1

,

b+1

,

c+1

is at least

2

, and their product is

12

.

(a+1)(b+1)(c+1) = 12 = 2 \times 2 \times 3

Two exponents are

1

and one exponent is

2

.

3List the candidates

#2 Make a Systematic List 6.EE.A.1

Put the exponent

2

on each prime in turn.

2^2 \times 5 \times 7 = 140, \quad 2 \times 5^2 \times 7 = 350, \quad 2 \times 5 \times 7^2 = 490

The largest three-digit one is

490

.

Answer: 490

4 · Reviewdoes it hold up?

$490 = 2 \times 5 \times 7^2$ : three different primes with $2 + 5 + 7 = 14$ , and $(1+1)(1+1)(2+1) = 12$ divisors. It has three digits.

Another way: To make the number as large as possible, give the square to the largest prime, $7$ , right away: $2 \times 5 \times 49 = 490$ .

Standardsmin grade 6

4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Recognizing which numbers are prime and choosing the three prime factors.
6.EE.A.1 Write and evaluate expressions with whole-number exponents — Writing and evaluating the numbers with exponents and counting divisors from the exponents.

💡Takeaway. The sum picks the primes; the divisor count picks the exponents.

Variant 3 medium

Find the largest three-digit natural number that satisfies both conditions below.

(a) Its prime factorization has exactly $3$ different prime factors, and these prime factors add up to $16$ .

(b) It has exactly $12$ divisors.

Your answer

Show solution

1 · Understandwhat's really being asked

A three-digit number has exactly three different prime factors that add up to $16$ , and it has exactly $12$ divisors. Find the largest such number.

Givens

Exactly $3$ different prime factors.
Those primes add up to $16$ .
Exactly $12$ divisors.
The number has three digits.

Unknowns

The largest such three-digit number.

Constraints

If $N = p^a \times q^b \times r^c$ , then $N$ has $(a+1)(b+1)(c+1)$ divisors.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #3 Eliminate Possibilities#7 Identify Subproblems

First pin down which three primes are possible, then use the divisor-count formula to find the exponents, and finally list every candidate and keep the largest three-digit one.

3 · Execute3 carry out the plan

1Find the three primes

#3 Eliminate Possibilities 4.OA.B.4

Three odd primes would add to an odd number, but

16

is even, so one prime is

2

. The other two add to

14

and must be different:

3

and

11

(

7 + 7

repeats a prime).

2 + 3 + 11 = 16

The prime factors must be

2

,

3

,

11

.

2Find the exponents

#7 Identify Subproblems 6.EE.A.1

Write

N = 2^a \times 3^b \times 11^c

with

a

,

b

,

c

at least

1

. Each factor

a+1

,

b+1

,

c+1

is at least

2

, and their product is

12

.

(a+1)(b+1)(c+1) = 12 = 2 \times 2 \times 3

Two exponents are

1

and one exponent is

2

.

3List the candidates

#2 Make a Systematic List 6.EE.A.1

Put the exponent

2

on each prime in turn.

2^2 \times 3 \times 11 = 132, \quad 2 \times 3^2 \times 11 = 198, \quad 2 \times 3 \times 11^2 = 726

The largest three-digit one is

726

.

Answer: 726

4 · Reviewdoes it hold up?

$726 = 2 \times 3 \times 11^2$ : three different primes with $2 + 3 + 11 = 16$ , and $(1+1)(1+1)(2+1) = 12$ divisors. It has three digits.

Another way: To make the number as large as possible, give the square to the largest prime, $11$ , right away: $2 \times 3 \times 121 = 726$ .

Standardsmin grade 6

4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Recognizing which numbers are prime and choosing the three prime factors.
6.EE.A.1 Write and evaluate expressions with whole-number exponents — Writing and evaluating the numbers with exponents and counting divisors from the exponents.

💡Takeaway. The sum picks the primes; the divisor count picks the exponents.

Variant 4 medium

Find the largest three-digit natural number that satisfies both conditions below.

(a) Its prime factorization has exactly $3$ different prime factors, and these prime factors add up to $16$ .

(b) It has exactly $16$ divisors.

Your answer

Show solution

1 · Understandwhat's really being asked

A three-digit number has exactly three different prime factors that add up to $16$ , and it has exactly $16$ divisors. Find the largest such number.

Givens

Exactly $3$ different prime factors.
Those primes add up to $16$ .
Exactly $16$ divisors.
The number has three digits.

Unknowns

The largest such three-digit number.

Constraints

If $N = p^a \times q^b \times r^c$ , then $N$ has $(a+1)(b+1)(c+1)$ divisors.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #3 Eliminate Possibilities#7 Identify Subproblems

First pin down which three primes are possible, then use the divisor-count formula to find the exponents, and finally list every candidate and keep the largest three-digit one.

3 · Execute3 carry out the plan

1Find the three primes

#3 Eliminate Possibilities 4.OA.B.4

Three odd primes would add to an odd number, but

16

is even, so one prime is

2

. The other two add to

14

and must be different:

3

and

11

(

7 + 7

repeats a prime).

2 + 3 + 11 = 16

The prime factors must be

2

,

3

,

11

.

2Find the exponents

#7 Identify Subproblems 6.EE.A.1

Write

N = 2^a \times 3^b \times 11^c

with

a

,

b

,

c

at least

1

. Each factor

a+1

,

b+1

,

c+1

is at least

2

, and their product is

16

.

(a+1)(b+1)(c+1) = 16 = 2 \times 2 \times 4

Two exponents are

1

and one exponent is

3

.

3List the candidates

#2 Make a Systematic List 6.EE.A.1

Put the exponent

3

on each prime in turn.

2^3 \times 3 \times 11 = 264, \quad 2 \times 3^3 \times 11 = 594, \quad 2 \times 3 \times 11^3 = 7986

The largest three-digit one is

594

.

Answer: 594

4 · Reviewdoes it hold up?

$594 = 2 \times 3^3 \times 11$ : three different primes with $2 + 3 + 11 = 16$ , and $(1+1)(3+1)(1+1) = 16$ divisors. It has three digits.

Another way: Order the candidates from the largest down: $7986$ , $594$ , $264$ . Skip any with more than three digits; the first one left is $594$ .

Standardsmin grade 6

4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Recognizing which numbers are prime and choosing the three prime factors.
6.EE.A.1 Write and evaluate expressions with whole-number exponents — Writing and evaluating the numbers with exponents and counting divisors from the exponents.

💡Takeaway. The sum picks the primes; the divisor count picks the exponents.

Variant 5 easy

Find the largest three-digit natural number that satisfies both conditions below.

(a) Its prime factorization has exactly $3$ different prime factors, and these prime factors add up to $12$ .

(b) It has exactly $16$ divisors.

Your answer

Show solution

1 · Understandwhat's really being asked

A three-digit number has exactly three different prime factors that add up to $12$ , and it has exactly $16$ divisors. Find the largest such number.

Givens

Exactly $3$ different prime factors.
Those primes add up to $12$ .
Exactly $16$ divisors.
The number has three digits.

Unknowns

The largest such three-digit number.

Constraints

If $N = p^a \times q^b \times r^c$ , then $N$ has $(a+1)(b+1)(c+1)$ divisors.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #3 Eliminate Possibilities#7 Identify Subproblems

First pin down which three primes are possible, then use the divisor-count formula to find the exponents, and finally list every candidate and keep the largest three-digit one.

3 · Execute3 carry out the plan

1Find the three primes

#3 Eliminate Possibilities 4.OA.B.4

Three odd primes would add to an odd number, but

12

is even, so one prime is

2

. The other two add to

10

and must be different:

3

and

7

(

5 + 5

repeats a prime).

2 + 3 + 7 = 12

The prime factors must be

2

,

3

,

7

.

2Find the exponents

#7 Identify Subproblems 6.EE.A.1

Write

N = 2^a \times 3^b \times 7^c

with

a

,

b

,

c

at least

1

. Each factor

a+1

,

b+1

,

c+1

is at least

2

, and their product is

16

.

(a+1)(b+1)(c+1) = 16 = 2 \times 2 \times 4

Two exponents are

1

and one exponent is

3

.

3List the candidates

#2 Make a Systematic List 6.EE.A.1

Put the exponent

3

on each prime in turn.

2^3 \times 3 \times 7 = 168, \quad 2 \times 3^3 \times 7 = 378, \quad 2 \times 3 \times 7^3 = 2058

The largest three-digit one is

378

.

Answer: 378

4 · Reviewdoes it hold up?

$378 = 2 \times 3^3 \times 7$ : three different primes with $2 + 3 + 7 = 12$ , and $(1+1)(3+1)(1+1) = 16$ divisors. It has three digits.

Another way: Order the candidates from the largest down: $2058$ , $378$ , $168$ . Skip any with more than three digits; the first one left is $378$ .

Standardsmin grade 6

4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Recognizing which numbers are prime and choosing the three prime factors.
6.EE.A.1 Write and evaluate expressions with whole-number exponents — Writing and evaluating the numbers with exponents and counting divisors from the exponents.

💡Takeaway. The sum picks the primes; the divisor count picks the exponents.

Variant 6 easy

Find the largest three-digit natural number that satisfies both conditions below.

(a) Its prime factorization has exactly $3$ different prime factors, and these prime factors add up to $10$ .

(b) It has exactly $16$ divisors.

Your answer

Show solution

1 · Understandwhat's really being asked

A three-digit number has exactly three different prime factors that add up to $10$ , and it has exactly $16$ divisors. Find the largest such number.

Givens

Exactly $3$ different prime factors.
Those primes add up to $10$ .
Exactly $16$ divisors.
The number has three digits.

Unknowns

The largest such three-digit number.

Constraints

If $N = p^a \times q^b \times r^c$ , then $N$ has $(a+1)(b+1)(c+1)$ divisors.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #3 Eliminate Possibilities#7 Identify Subproblems

First pin down which three primes are possible, then use the divisor-count formula to find the exponents, and finally list every candidate and keep the largest three-digit one.

3 · Execute3 carry out the plan

1Find the three primes

#3 Eliminate Possibilities 4.OA.B.4

Three odd primes would add to an odd number, but

10

is even, so one prime is

2

. The other two add to

8

and must be different:

3

and

5

.

2 + 3 + 5 = 10

The prime factors must be

2

,

3

,

5

.

2Find the exponents

#7 Identify Subproblems 6.EE.A.1

Write

N = 2^a \times 3^b \times 5^c

with

a

,

b

,

c

at least

1

. Each factor

a+1

,

b+1

,

c+1

is at least

2

, and their product is

16

.

(a+1)(b+1)(c+1) = 16 = 2 \times 2 \times 4

Two exponents are

1

and one exponent is

3

.

3List the candidates

#2 Make a Systematic List 6.EE.A.1

Put the exponent

3

on each prime in turn.

2^3 \times 3 \times 5 = 120, \quad 2 \times 3^3 \times 5 = 270, \quad 2 \times 3 \times 5^3 = 750

The largest three-digit one is

750

.

Answer: 750

4 · Reviewdoes it hold up?

$750 = 2 \times 3 \times 5^3$ : three different primes with $2 + 3 + 5 = 10$ , and $(1+1)(1+1)(3+1) = 16$ divisors. It has three digits.

Another way: To make the number as large as possible, give the exponent $3$ to the largest prime, $5$ , right away: $2 \times 3 \times 125 = 750$ .

Standardsmin grade 6

4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Recognizing which numbers are prime and choosing the three prime factors.
6.EE.A.1 Write and evaluate expressions with whole-number exponents — Writing and evaluating the numbers with exponents and counting divisors from the exponents.

💡Takeaway. The sum picks the primes; the divisor count picks the exponents.

Variant 7 medium

Find the largest three-digit natural number that satisfies both conditions below.

(a) Its prime factorization has exactly $3$ different prime factors, and these prime factors add up to $10$ .

(b) It has exactly $18$ divisors.

Your answer

Show solution

1 · Understandwhat's really being asked

A three-digit number has exactly three different prime factors that add up to $10$ , and it has exactly $18$ divisors. Find the largest such number.

Givens

Exactly $3$ different prime factors.
Those primes add up to $10$ .
Exactly $18$ divisors.
The number has three digits.

Unknowns

The largest such three-digit number.

Constraints

If $N = p^a \times q^b \times r^c$ , then $N$ has $(a+1)(b+1)(c+1)$ divisors.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #3 Eliminate Possibilities#7 Identify Subproblems

First pin down which three primes are possible, then use the divisor-count formula to find the exponents, and finally list every candidate and keep the largest three-digit one.

3 · Execute3 carry out the plan

1Find the three primes

#3 Eliminate Possibilities 4.OA.B.4

Three odd primes would add to an odd number, but

10

is even, so one prime is

2

. The other two add to

8

and must be different:

3

and

5

.

2 + 3 + 5 = 10

The prime factors must be

2

,

3

,

5

.

2Find the exponents

#7 Identify Subproblems 6.EE.A.1

Write

N = 2^a \times 3^b \times 5^c

with

a

,

b

,

c

at least

1

. Each factor

a+1

,

b+1

,

c+1

is at least

2

, and their product is

18

.

(a+1)(b+1)(c+1) = 18 = 2 \times 3 \times 3

One exponent is

1

and two exponents are

2

.

3List the candidates

#2 Make a Systematic List 6.EE.A.1

Put the exponent

1

on each prime in turn; the other two get

2

.

2 \times 3^2 \times 5^2 = 450, \quad 2^2 \times 3 \times 5^2 = 300, \quad 2^2 \times 3^2 \times 5 = 180

The largest three-digit one is

450

.

Answer: 450

4 · Reviewdoes it hold up?

$450 = 2 \times 3^2 \times 5^2$ : three different primes with $2 + 3 + 5 = 10$ , and $(1+1)(2+1)(2+1) = 18$ divisors. It has three digits.

Another way: To make the number as large as possible, give the exponent $1$ to the smallest prime, $2$ , right away: $2 \times 9 \times 25 = 450$ .

Standardsmin grade 6

4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Recognizing which numbers are prime and choosing the three prime factors.
6.EE.A.1 Write and evaluate expressions with whole-number exponents — Writing and evaluating the numbers with exponents and counting divisors from the exponents.

💡Takeaway. The sum picks the primes; the divisor count picks the exponents.

Variant 8 hard

Find the largest three-digit natural number that satisfies both conditions below.

(a) Its prime factorization has exactly $3$ different prime factors, and these prime factors add up to $16$ .

(b) It has exactly $18$ divisors.

Your answer

Show solution

1 · Understandwhat's really being asked

A three-digit number has exactly three different prime factors that add up to $16$ , and it has exactly $18$ divisors. Find the largest such number.

Givens

Exactly $3$ different prime factors.
Those primes add up to $16$ .
Exactly $18$ divisors.
The number has three digits.

Unknowns

The largest such three-digit number.

Constraints

If $N = p^a \times q^b \times r^c$ , then $N$ has $(a+1)(b+1)(c+1)$ divisors.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #3 Eliminate Possibilities#7 Identify Subproblems

First pin down which three primes are possible, then use the divisor-count formula to find the exponents, and finally list every candidate and keep the largest three-digit one.

3 · Execute3 carry out the plan

1Find the three primes

#3 Eliminate Possibilities 4.OA.B.4

Three odd primes would add to an odd number, but

16

is even, so one prime is

2

. The other two add to

14

and must be different:

3

and

11

(

7 + 7

repeats a prime).

2 + 3 + 11 = 16

The prime factors must be

2

,

3

,

11

.

2Find the exponents

#7 Identify Subproblems 6.EE.A.1

Write

N = 2^a \times 3^b \times 11^c

with

a

,

b

,

c

at least

1

. Each factor

a+1

,

b+1

,

c+1

is at least

2

, and their product is

18

.

(a+1)(b+1)(c+1) = 18 = 2 \times 3 \times 3

One exponent is

1

and two exponents are

2

.

3List the candidates

#2 Make a Systematic List 6.EE.A.1

Put the exponent

1

on each prime in turn; the other two get

2

.

2 \times 3^2 \times 11^2 = 2178, \quad 2^2 \times 3 \times 11^2 = 1452, \quad 2^2 \times 3^2 \times 11 = 396

The largest three-digit one is

396

.

Answer: 396

4 · Reviewdoes it hold up?

$396 = 2^2 \times 3^2 \times 11$ : three different primes with $2 + 3 + 11 = 16$ , and $(2+1)(2+1)(1+1) = 18$ divisors. It has three digits.

Another way: Order the candidates from the largest down: $2178$ , $1452$ , $396$ . Skip any with more than three digits; the first one left is $396$ .

Standardsmin grade 6

4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Recognizing which numbers are prime and choosing the three prime factors.
6.EE.A.1 Write and evaluate expressions with whole-number exponents — Writing and evaluating the numbers with exponents and counting divisors from the exponents.

💡Takeaway. The sum picks the primes; the divisor count picks the exponents.

Variant 9 hard

Find the largest three-digit natural number that satisfies both conditions below.

(a) Its prime factorization has exactly $3$ different prime factors, and these prime factors add up to $14$ .

(b) It has exactly $18$ divisors.

Your answer

Show solution

1 · Understandwhat's really being asked

A three-digit number has exactly three different prime factors that add up to $14$ , and it has exactly $18$ divisors. Find the largest such number.

Givens

Exactly $3$ different prime factors.
Those primes add up to $14$ .
Exactly $18$ divisors.
The number has three digits.

Unknowns

The largest such three-digit number.

Constraints

If $N = p^a \times q^b \times r^c$ , then $N$ has $(a+1)(b+1)(c+1)$ divisors.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #3 Eliminate Possibilities#7 Identify Subproblems

First pin down which three primes are possible, then use the divisor-count formula to find the exponents, and finally list every candidate and keep the largest three-digit one.

3 · Execute3 carry out the plan

1Find the three primes

#3 Eliminate Possibilities 4.OA.B.4

Three odd primes would add to an odd number, but

14

is even, so one prime is

2

. The other two add to

12

and must be different:

5

and

7

.

2 + 5 + 7 = 14

The prime factors must be

2

,

5

,

7

.

2Find the exponents

#7 Identify Subproblems 6.EE.A.1

Write

N = 2^a \times 5^b \times 7^c

with

a

,

b

,

c

at least

1

. Each factor

a+1

,

b+1

,

c+1

is at least

2

, and their product is

18

.

(a+1)(b+1)(c+1) = 18 = 2 \times 3 \times 3

One exponent is

1

and two exponents are

2

.

3List the candidates

#2 Make a Systematic List 6.EE.A.1

Put the exponent

1

on each prime in turn; the other two get

2

.

2 \times 5^2 \times 7^2 = 2450, \quad 2^2 \times 5 \times 7^2 = 980, \quad 2^2 \times 5^2 \times 7 = 700

The largest three-digit one is

980

.

Answer: 980

4 · Reviewdoes it hold up?

$980 = 2^2 \times 5 \times 7^2$ : three different primes with $2 + 5 + 7 = 14$ , and $(2+1)(1+1)(2+1) = 18$ divisors. It has three digits.

Another way: Order the candidates from the largest down: $2450$ , $980$ , $700$ . Skip any with more than three digits; the first one left is $980$ .

Standardsmin grade 6

4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Recognizing which numbers are prime and choosing the three prime factors.
6.EE.A.1 Write and evaluate expressions with whole-number exponents — Writing and evaluating the numbers with exponents and counting divisors from the exponents.

💡Takeaway. The sum picks the primes; the divisor count picks the exponents.

Variant 10 medium

Find the largest three-digit natural number that satisfies both conditions below.

(a) Its prime factorization has exactly $3$ different prime factors, and these prime factors add up to $12$ .

(b) It has exactly $18$ divisors.

Your answer

Show solution

1 · Understandwhat's really being asked

A three-digit number has exactly three different prime factors that add up to $12$ , and it has exactly $18$ divisors. Find the largest such number.

Givens

Exactly $3$ different prime factors.
Those primes add up to $12$ .
Exactly $18$ divisors.
The number has three digits.

Unknowns

The largest such three-digit number.

Constraints

If $N = p^a \times q^b \times r^c$ , then $N$ has $(a+1)(b+1)(c+1)$ divisors.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #3 Eliminate Possibilities#7 Identify Subproblems

First pin down which three primes are possible, then use the divisor-count formula to find the exponents, and finally list every candidate and keep the largest three-digit one.

3 · Execute3 carry out the plan

1Find the three primes

#3 Eliminate Possibilities 4.OA.B.4

Three odd primes would add to an odd number, but

12

is even, so one prime is

2

. The other two add to

10

and must be different:

3

and

7

(

5 + 5

repeats a prime).

2 + 3 + 7 = 12

The prime factors must be

2

,

3

,

7

.

2Find the exponents

#7 Identify Subproblems 6.EE.A.1

Write

N = 2^a \times 3^b \times 7^c

with

a

,

b

,

c

at least

1

. Each factor

a+1

,

b+1

,

c+1

is at least

2

, and their product is

18

.

(a+1)(b+1)(c+1) = 18 = 2 \times 3 \times 3

One exponent is

1

and two exponents are

2

.

3List the candidates

#2 Make a Systematic List 6.EE.A.1

Put the exponent

1

on each prime in turn; the other two get

2

.

2 \times 3^2 \times 7^2 = 882, \quad 2^2 \times 3 \times 7^2 = 588, \quad 2^2 \times 3^2 \times 7 = 252

The largest three-digit one is

882

.

Answer: 882

4 · Reviewdoes it hold up?

$882 = 2 \times 3^2 \times 7^2$ : three different primes with $2 + 3 + 7 = 12$ , and $(1+1)(2+1)(2+1) = 18$ divisors. It has three digits.

Another way: To make the number as large as possible, give the exponent $1$ to the smallest prime, $2$ , right away: $2 \times 9 \times 49 = 882$ .

Standardsmin grade 6

4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Recognizing which numbers are prime and choosing the three prime factors.
6.EE.A.1 Write and evaluate expressions with whole-number exponents — Writing and evaluating the numbers with exponents and counting divisors from the exponents.

💡Takeaway. The sum picks the primes; the divisor count picks the exponents.

Variant 11 hard

Find the largest three-digit natural number that satisfies both conditions below.

(a) Its prime factorization has exactly $3$ different prime factors, and these prime factors add up to $12$ .

(b) It has exactly $20$ divisors.

Your answer

Show solution

1 · Understandwhat's really being asked

A three-digit number has exactly three different prime factors that add up to $12$ , and it has exactly $20$ divisors. Find the largest such number.

Givens

Exactly $3$ different prime factors.
Those primes add up to $12$ .
Exactly $20$ divisors.
The number has three digits.

Unknowns

The largest such three-digit number.

Constraints

If $N = p^a \times q^b \times r^c$ , then $N$ has $(a+1)(b+1)(c+1)$ divisors.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #3 Eliminate Possibilities#7 Identify Subproblems

First pin down which three primes are possible, then use the divisor-count formula to find the exponents, and finally list every candidate and keep the largest three-digit one.

3 · Execute3 carry out the plan

1Find the three primes

#3 Eliminate Possibilities 4.OA.B.4

Three odd primes would add to an odd number, but

12

is even, so one prime is

2

. The other two add to

10

and must be different:

3

and

7

(

5 + 5

repeats a prime).

2 + 3 + 7 = 12

The prime factors must be

2

,

3

,

7

.

2Find the exponents

#7 Identify Subproblems 6.EE.A.1

Write

N = 2^a \times 3^b \times 7^c

with

a

,

b

,

c

at least

1

. Each factor

a+1

,

b+1

,

c+1

is at least

2

, and their product is

20

.

(a+1)(b+1)(c+1) = 20 = 2 \times 2 \times 5

Two exponents are

1

and one exponent is

4

.

3List the candidates

#2 Make a Systematic List 6.EE.A.1

Put the exponent

4

on each prime in turn.

2^4 \times 3 \times 7 = 336, \quad 2 \times 3^4 \times 7 = 1134, \quad 2 \times 3 \times 7^4 = 14406

The largest three-digit one is

336

.

Answer: 336

4 · Reviewdoes it hold up?

$336 = 2^4 \times 3 \times 7$ : three different primes with $2 + 3 + 7 = 12$ , and $(4+1)(1+1)(1+1) = 20$ divisors. It has three digits.

Another way: Order the candidates from the largest down: $14406$ , $1134$ , $336$ . Skip any with more than three digits; the first one left is $336$ .

Standardsmin grade 6

4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Recognizing which numbers are prime and choosing the three prime factors.
6.EE.A.1 Write and evaluate expressions with whole-number exponents — Writing and evaluating the numbers with exponents and counting divisors from the exponents.

💡Takeaway. The sum picks the primes; the divisor count picks the exponents.

Variant 12 hard

Find the largest three-digit natural number that satisfies both conditions below.

(a) Its prime factorization has exactly $3$ different prime factors, and these prime factors add up to $10$ .

(b) It has exactly $20$ divisors.

Your answer

Show solution

1 · Understandwhat's really being asked

A three-digit number has exactly three different prime factors that add up to $10$ , and it has exactly $20$ divisors. Find the largest such number.

Givens

Exactly $3$ different prime factors.
Those primes add up to $10$ .
Exactly $20$ divisors.
The number has three digits.

Unknowns

The largest such three-digit number.

Constraints

If $N = p^a \times q^b \times r^c$ , then $N$ has $(a+1)(b+1)(c+1)$ divisors.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #3 Eliminate Possibilities#7 Identify Subproblems

First pin down which three primes are possible, then use the divisor-count formula to find the exponents, and finally list every candidate and keep the largest three-digit one.

3 · Execute3 carry out the plan

1Find the three primes

#3 Eliminate Possibilities 4.OA.B.4

Three odd primes would add to an odd number, but

10

is even, so one prime is

2

. The other two add to

8

and must be different:

3

and

5

.

2 + 3 + 5 = 10

The prime factors must be

2

,

3

,

5

.

2Find the exponents

#7 Identify Subproblems 6.EE.A.1

Write

N = 2^a \times 3^b \times 5^c

with

a

,

b

,

c

at least

1

. Each factor

a+1

,

b+1

,

c+1

is at least

2

, and their product is

20

.

(a+1)(b+1)(c+1) = 20 = 2 \times 2 \times 5

Two exponents are

1

and one exponent is

4

.

3List the candidates

#2 Make a Systematic List 6.EE.A.1

Put the exponent

4

on each prime in turn.

2^4 \times 3 \times 5 = 240, \quad 2 \times 3^4 \times 5 = 810, \quad 2 \times 3 \times 5^4 = 3750

The largest three-digit one is

810

.

Answer: 810

4 · Reviewdoes it hold up?

$810 = 2 \times 3^4 \times 5$ : three different primes with $2 + 3 + 5 = 10$ , and $(1+1)(4+1)(1+1) = 20$ divisors. It has three digits.

Another way: Order the candidates from the largest down: $3750$ , $810$ , $240$ . Skip any with more than three digits; the first one left is $810$ .

Standardsmin grade 6

4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Recognizing which numbers are prime and choosing the three prime factors.
6.EE.A.1 Write and evaluate expressions with whole-number exponents — Writing and evaluating the numbers with exponents and counting divisors from the exponents.

💡Takeaway. The sum picks the primes; the divisor count picks the exponents.

Fix primes by sum, exponents by divisor count · 12 practice problems

Generated variants — 12

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