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Deduce missing digits in a multiplication · 12 practice problems

3.OA.C.73.NBT.A.3

Generated variants — 12

Freshly produced from the archetype’s parameters — problem, figure, and solution derived together.

Variant 1 easy answer: A = 1, B = 5

In the multiplication below, find the digits $A$ and $B$ .

$A47 \times B = 735$

Here $A47$ is a three-digit number whose hundreds digit is $A$ , tens digit is $4$ , and ones digit is $7$ , and $B$ is a single digit.

Show solution

1 · Understandwhat's really being asked

A three-digit number A47 (hundreds digit A, tens 4, ones 7) times a single digit B equals 735. Find A and B.

Givens

A47 x B = 735.
A47 has tens digit 4 and ones digit 7; A is its hundreds digit.
B is a single digit.

Unknowns

The digit A and the digit B.

Constraints

A is a digit (1 through 9, since A is the hundreds digit of a three-digit number).
B is a single digit (1 through 9).

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #6 Guess and Check

Start at the ones place: 7 times B must end in 5, which only a couple of digits do. Test each candidate B by dividing 735 to see if it rebuilds a clean A47 form.

3 · Execute3 carry out the plan

1Use the ones digit to limit B

#2 Make a Systematic List 3.OA.C.7

The product ends in 5, and the ones digit of A47 is 7, so 7 x B must end in 5. Checking the digits, only B = 5 works for the ones place.

7 \times 5 = 35

The ones digit of a product depends only on the ones digits of the factors.

2Test each candidate B

#6 Guess and Check 3.OA.C.7

Divide the product by each candidate: if B = 5, then 735 / 5 = 147. Only B = 5 rebuilds the form A47 with tens 4 and ones 7, giving A47 = 147, so A = 1.

735 \div 5 = 147

Dividing the product by B should rebuild the original number with the right tens and ones.

3Confirm

#6 Guess and Check 3.NBT.A.3

Check 147 x 5 by place value: 100 x 5 = 500, 40 x 5 = 200, 7 x 5 = 35, total 735. So A = 1 and B = 5.

147 \times 5 = 100 \times 5 = 500, 40 \times 5 = 200, 7 \times 5 = 35 \Rightarrow 735

Multiplying back must return the original product.

Answer: A = 1, B = 5

4 · Reviewdoes it hold up?

147 x 5 = 735 matches exactly, the tens digit is 4 and ones digit is 7 as required, so the solution fits all conditions.

Another way: Estimate B: 735 divided by about 147 is roughly 5, pointing straight to B = 5, then A47 = 147 gives A = 1.

Standardsmin grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Reasoning about ones digits and checking the products.
3.NBT.A.3 Multiply one-digit whole numbers by multiples of 10 — Multiplying 147 by 5 using place value to confirm.

💡Takeaway. Look at the ones digit first to narrow B, then divide to check -- classic Grade 3 detective work!

Variant 2 easy answer: A = 4, B = 2

In the multiplication below, find the digits $A$ and $B$ .

$A57 \times B = 914$

Here $A57$ is a three-digit number whose hundreds digit is $A$ , tens digit is $5$ , and ones digit is $7$ , and $B$ is a single digit.

Show solution

1 · Understandwhat's really being asked

A three-digit number A57 (hundreds digit A, tens 5, ones 7) times a single digit B equals 914. Find A and B.

Givens

A57 x B = 914.
A57 has tens digit 5 and ones digit 7; A is its hundreds digit.
B is a single digit.

Unknowns

The digit A and the digit B.

Constraints

A is a digit (1 through 9, since A is the hundreds digit of a three-digit number).
B is a single digit (1 through 9).

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #6 Guess and Check

Start at the ones place: 7 times B must end in 4, which only a couple of digits do. Test each candidate B by dividing 914 to see if it rebuilds a clean A57 form.

3 · Execute3 carry out the plan

1Use the ones digit to limit B

#2 Make a Systematic List 3.OA.C.7

The product ends in 4, and the ones digit of A57 is 7, so 7 x B must end in 4. Checking the digits, only B = 2 works for the ones place.

7 \times 2 = 14

The ones digit of a product depends only on the ones digits of the factors.

2Test each candidate B

#6 Guess and Check 3.OA.C.7

Divide the product by each candidate: if B = 2, then 914 / 2 = 457. Only B = 2 rebuilds the form A57 with tens 5 and ones 7, giving A57 = 457, so A = 4.

914 \div 2 = 457

Dividing the product by B should rebuild the original number with the right tens and ones.

3Confirm

#6 Guess and Check 3.NBT.A.3

Check 457 x 2 by place value: 400 x 2 = 800, 50 x 2 = 100, 7 x 2 = 14, total 914. So A = 4 and B = 2.

457 \times 2 = 400 \times 2 = 800, 50 \times 2 = 100, 7 \times 2 = 14 \Rightarrow 914

Multiplying back must return the original product.

Answer: A = 4, B = 2

4 · Reviewdoes it hold up?

457 x 2 = 914 matches exactly, the tens digit is 5 and ones digit is 7 as required, so the solution fits all conditions.

Another way: Estimate B: 914 divided by about 457 is roughly 2, pointing straight to B = 2, then A57 = 457 gives A = 4.

Standardsmin grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Reasoning about ones digits and checking the products.
3.NBT.A.3 Multiply one-digit whole numbers by multiples of 10 — Multiplying 457 by 2 using place value to confirm.

💡Takeaway. Look at the ones digit first to narrow B, then divide to check -- classic Grade 3 detective work!

Variant 3 easy answer: A = 3, B = 3

In the multiplication below, find the digits $A$ and $B$ .

$A26 \times B = 978$

Here $A26$ is a three-digit number whose hundreds digit is $A$ , tens digit is $2$ , and ones digit is $6$ , and $B$ is a single digit.

Show solution

1 · Understandwhat's really being asked

A three-digit number A26 (hundreds digit A, tens 2, ones 6) times a single digit B equals 978. Find A and B.

Givens

A26 x B = 978.
A26 has tens digit 2 and ones digit 6; A is its hundreds digit.
B is a single digit.

Unknowns

The digit A and the digit B.

Constraints

A is a digit (1 through 9, since A is the hundreds digit of a three-digit number).
B is a single digit (1 through 9).

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #6 Guess and Check

Start at the ones place: 6 times B must end in 8, which only a couple of digits do. Test each candidate B by dividing 978 to see if it rebuilds a clean A26 form.

3 · Execute3 carry out the plan

1Use the ones digit to limit B

#2 Make a Systematic List 3.OA.C.7

The product ends in 8, and the ones digit of A26 is 6, so 6 x B must end in 8. Checking the digits, only B = 3 or 8 works for the ones place.

6 \times 3 = 18,\quad 6 \times 8 = 48

The ones digit of a product depends only on the ones digits of the factors.

2Test each candidate B

#6 Guess and Check 3.OA.C.7

Divide the product by each candidate: if B = 3, then 978 / 3 = 326; if B = 8, then 978 / 8 is not a clean three-digit number. Only B = 3 rebuilds the form A26 with tens 2 and ones 6, giving A26 = 326, so A = 3.

978 \div 3 = 326

Dividing the product by B should rebuild the original number with the right tens and ones.

3Confirm

#6 Guess and Check 3.NBT.A.3

Check 326 x 3 by place value: 300 x 3 = 900, 20 x 3 = 60, 6 x 3 = 18, total 978. So A = 3 and B = 3.

326 \times 3 = 300 \times 3 = 900, 20 \times 3 = 60, 6 \times 3 = 18 \Rightarrow 978

Multiplying back must return the original product.

Answer: A = 3, B = 3

4 · Reviewdoes it hold up?

326 x 3 = 978 matches exactly, the tens digit is 2 and ones digit is 6 as required, so the solution fits all conditions.

Another way: Estimate B: 978 divided by about 326 is roughly 3, pointing straight to B = 3, then A26 = 326 gives A = 3.

Standardsmin grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Reasoning about ones digits and checking the products.
3.NBT.A.3 Multiply one-digit whole numbers by multiples of 10 — Multiplying 326 by 3 using place value to confirm.

💡Takeaway. Look at the ones digit first to narrow B, then divide to check -- classic Grade 3 detective work!

Variant 4 easy answer: A = 2, B = 4

In the multiplication below, find the digits $A$ and $B$ .

$A53 \times B = 1012$

Here $A53$ is a three-digit number whose hundreds digit is $A$ , tens digit is $5$ , and ones digit is $3$ , and $B$ is a single digit.

Show solution

1 · Understandwhat's really being asked

A three-digit number A53 (hundreds digit A, tens 5, ones 3) times a single digit B equals 1012. Find A and B.

Givens

A53 x B = 1012.
A53 has tens digit 5 and ones digit 3; A is its hundreds digit.
B is a single digit.

Unknowns

The digit A and the digit B.

Constraints

A is a digit (1 through 9, since A is the hundreds digit of a three-digit number).
B is a single digit (1 through 9).

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #6 Guess and Check

Start at the ones place: 3 times B must end in 2, which only a couple of digits do. Test each candidate B by dividing 1012 to see if it rebuilds a clean A53 form.

3 · Execute3 carry out the plan

1Use the ones digit to limit B

#2 Make a Systematic List 3.OA.C.7

The product ends in 2, and the ones digit of A53 is 3, so 3 x B must end in 2. Checking the digits, only B = 4 works for the ones place.

3 \times 4 = 12

The ones digit of a product depends only on the ones digits of the factors.

2Test each candidate B

#6 Guess and Check 3.OA.C.7

Divide the product by each candidate: if B = 4, then 1012 / 4 = 253. Only B = 4 rebuilds the form A53 with tens 5 and ones 3, giving A53 = 253, so A = 2.

1012 \div 4 = 253

Dividing the product by B should rebuild the original number with the right tens and ones.

3Confirm

#6 Guess and Check 3.NBT.A.3

Check 253 x 4 by place value: 200 x 4 = 800, 50 x 4 = 200, 3 x 4 = 12, total 1012. So A = 2 and B = 4.

253 \times 4 = 200 \times 4 = 800, 50 \times 4 = 200, 3 \times 4 = 12 \Rightarrow 1012

Multiplying back must return the original product.

Answer: A = 2, B = 4

4 · Reviewdoes it hold up?

253 x 4 = 1012 matches exactly, the tens digit is 5 and ones digit is 3 as required, so the solution fits all conditions.

Another way: Estimate B: 1012 divided by about 253 is roughly 4, pointing straight to B = 4, then A53 = 253 gives A = 2.

Standardsmin grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Reasoning about ones digits and checking the products.
3.NBT.A.3 Multiply one-digit whole numbers by multiples of 10 — Multiplying 253 by 4 using place value to confirm.

💡Takeaway. Look at the ones digit first to narrow B, then divide to check -- classic Grade 3 detective work!

Variant 5 medium answer: A = 1, B = 6

In the multiplication below, find the digits $A$ and $B$ .

$A94 \times B = 1164$

Here $A94$ is a three-digit number whose hundreds digit is $A$ , tens digit is $9$ , and ones digit is $4$ , and $B$ is a single digit.

Show solution

1 · Understandwhat's really being asked

A three-digit number A94 (hundreds digit A, tens 9, ones 4) times a single digit B equals 1164. Find A and B.

Givens

A94 x B = 1164.
A94 has tens digit 9 and ones digit 4; A is its hundreds digit.
B is a single digit.

Unknowns

The digit A and the digit B.

Constraints

A is a digit (1 through 9, since A is the hundreds digit of a three-digit number).
B is a single digit (1 through 9).

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #6 Guess and Check

Start at the ones place: 4 times B must end in 4, which only a couple of digits do. Test each candidate B by dividing 1164 to see if it rebuilds a clean A94 form.

3 · Execute3 carry out the plan

1Use the ones digit to limit B

#2 Make a Systematic List 3.OA.C.7

The product ends in 4, and the ones digit of A94 is 4, so 4 x B must end in 4. Checking the digits, only B = 1 or 6 works for the ones place.

4 \times 1 = 4,\quad 4 \times 6 = 24

The ones digit of a product depends only on the ones digits of the factors.

2Test each candidate B

#6 Guess and Check 3.OA.C.7

Divide the product by each candidate: if B = 1, then 1164 / 1 is not a clean three-digit number; if B = 6, then 1164 / 6 = 194. Only B = 6 rebuilds the form A94 with tens 9 and ones 4, giving A94 = 194, so A = 1.

1164 \div 6 = 194

Dividing the product by B should rebuild the original number with the right tens and ones.

3Confirm

#6 Guess and Check 3.NBT.A.3

Check 194 x 6 by place value: 100 x 6 = 600, 90 x 6 = 540, 4 x 6 = 24, total 1164. So A = 1 and B = 6.

194 \times 6 = 100 \times 6 = 600, 90 \times 6 = 540, 4 \times 6 = 24 \Rightarrow 1164

Multiplying back must return the original product.

Answer: A = 1, B = 6

4 · Reviewdoes it hold up?

194 x 6 = 1164 matches exactly, the tens digit is 9 and ones digit is 4 as required, so the solution fits all conditions.

Another way: Estimate B: 1164 divided by about 194 is roughly 6, pointing straight to B = 6, then A94 = 194 gives A = 1.

Standardsmin grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Reasoning about ones digits and checking the products.
3.NBT.A.3 Multiply one-digit whole numbers by multiples of 10 — Multiplying 194 by 6 using place value to confirm.

💡Takeaway. Look at the ones digit first to narrow B, then divide to check -- classic Grade 3 detective work!

Variant 6 medium answer: A = 4, B = 3

In the multiplication below, find the digits $A$ and $B$ .

$A12 \times B = 1236$

Here $A12$ is a three-digit number whose hundreds digit is $A$ , tens digit is $1$ , and ones digit is $2$ , and $B$ is a single digit.

Show solution

1 · Understandwhat's really being asked

A three-digit number A12 (hundreds digit A, tens 1, ones 2) times a single digit B equals 1236. Find A and B.

Givens

A12 x B = 1236.
A12 has tens digit 1 and ones digit 2; A is its hundreds digit.
B is a single digit.

Unknowns

The digit A and the digit B.

Constraints

A is a digit (1 through 9, since A is the hundreds digit of a three-digit number).
B is a single digit (1 through 9).

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #6 Guess and Check

Start at the ones place: 2 times B must end in 6, which only a couple of digits do. Test each candidate B by dividing 1236 to see if it rebuilds a clean A12 form.

3 · Execute3 carry out the plan

1Use the ones digit to limit B

#2 Make a Systematic List 3.OA.C.7

The product ends in 6, and the ones digit of A12 is 2, so 2 x B must end in 6. Checking the digits, only B = 3 or 8 works for the ones place.

2 \times 3 = 6,\quad 2 \times 8 = 16

The ones digit of a product depends only on the ones digits of the factors.

2Test each candidate B

#6 Guess and Check 3.OA.C.7

Divide the product by each candidate: if B = 3, then 1236 / 3 = 412; if B = 8, then 1236 / 8 is not a clean three-digit number. Only B = 3 rebuilds the form A12 with tens 1 and ones 2, giving A12 = 412, so A = 4.

1236 \div 3 = 412

Dividing the product by B should rebuild the original number with the right tens and ones.

3Confirm

#6 Guess and Check 3.NBT.A.3

Check 412 x 3 by place value: 400 x 3 = 1200, 10 x 3 = 30, 2 x 3 = 6, total 1236. So A = 4 and B = 3.

412 \times 3 = 400 \times 3 = 1200, 10 \times 3 = 30, 2 \times 3 = 6 \Rightarrow 1236

Multiplying back must return the original product.

Answer: A = 4, B = 3

4 · Reviewdoes it hold up?

412 x 3 = 1236 matches exactly, the tens digit is 1 and ones digit is 2 as required, so the solution fits all conditions.

Another way: Estimate B: 1236 divided by about 412 is roughly 3, pointing straight to B = 3, then A12 = 412 gives A = 4.

Standardsmin grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Reasoning about ones digits and checking the products.
3.NBT.A.3 Multiply one-digit whole numbers by multiples of 10 — Multiplying 412 by 3 using place value to confirm.

💡Takeaway. Look at the ones digit first to narrow B, then divide to check -- classic Grade 3 detective work!

Variant 7 medium answer: A = 1, B = 7

In the multiplication below, find the digits $A$ and $B$ .

$A83 \times B = 1281$

Here $A83$ is a three-digit number whose hundreds digit is $A$ , tens digit is $8$ , and ones digit is $3$ , and $B$ is a single digit.

Show solution

1 · Understandwhat's really being asked

A three-digit number A83 (hundreds digit A, tens 8, ones 3) times a single digit B equals 1281. Find A and B.

Givens

A83 x B = 1281.
A83 has tens digit 8 and ones digit 3; A is its hundreds digit.
B is a single digit.

Unknowns

The digit A and the digit B.

Constraints

A is a digit (1 through 9, since A is the hundreds digit of a three-digit number).
B is a single digit (1 through 9).

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #6 Guess and Check

Start at the ones place: 3 times B must end in 1, which only a couple of digits do. Test each candidate B by dividing 1281 to see if it rebuilds a clean A83 form.

3 · Execute3 carry out the plan

1Use the ones digit to limit B

#2 Make a Systematic List 3.OA.C.7

The product ends in 1, and the ones digit of A83 is 3, so 3 x B must end in 1. Checking the digits, only B = 7 works for the ones place.

3 \times 7 = 21

The ones digit of a product depends only on the ones digits of the factors.

2Test each candidate B

#6 Guess and Check 3.OA.C.7

Divide the product by each candidate: if B = 7, then 1281 / 7 = 183. Only B = 7 rebuilds the form A83 with tens 8 and ones 3, giving A83 = 183, so A = 1.

1281 \div 7 = 183

Dividing the product by B should rebuild the original number with the right tens and ones.

3Confirm

#6 Guess and Check 3.NBT.A.3

Check 183 x 7 by place value: 100 x 7 = 700, 80 x 7 = 560, 3 x 7 = 21, total 1281. So A = 1 and B = 7.

183 \times 7 = 100 \times 7 = 700, 80 \times 7 = 560, 3 \times 7 = 21 \Rightarrow 1281

Multiplying back must return the original product.

Answer: A = 1, B = 7

4 · Reviewdoes it hold up?

183 x 7 = 1281 matches exactly, the tens digit is 8 and ones digit is 3 as required, so the solution fits all conditions.

Another way: Estimate B: 1281 divided by about 183 is roughly 7, pointing straight to B = 7, then A83 = 183 gives A = 1.

Standardsmin grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Reasoning about ones digits and checking the products.
3.NBT.A.3 Multiply one-digit whole numbers by multiples of 10 — Multiplying 183 by 7 using place value to confirm.

💡Takeaway. Look at the ones digit first to narrow B, then divide to check -- classic Grade 3 detective work!

Variant 8 medium answer: A = 3, B = 4

In the multiplication below, find the digits $A$ and $B$ .

$A76 \times B = 1504$

Here $A76$ is a three-digit number whose hundreds digit is $A$ , tens digit is $7$ , and ones digit is $6$ , and $B$ is a single digit.

Show solution

1 · Understandwhat's really being asked

A three-digit number A76 (hundreds digit A, tens 7, ones 6) times a single digit B equals 1504. Find A and B.

Givens

A76 x B = 1504.
A76 has tens digit 7 and ones digit 6; A is its hundreds digit.
B is a single digit.

Unknowns

The digit A and the digit B.

Constraints

A is a digit (1 through 9, since A is the hundreds digit of a three-digit number).
B is a single digit (1 through 9).

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #6 Guess and Check

Start at the ones place: 6 times B must end in 4, which only a couple of digits do. Test each candidate B by dividing 1504 to see if it rebuilds a clean A76 form.

3 · Execute3 carry out the plan

1Use the ones digit to limit B

#2 Make a Systematic List 3.OA.C.7

The product ends in 4, and the ones digit of A76 is 6, so 6 x B must end in 4. Checking the digits, only B = 4 or 9 works for the ones place.

6 \times 4 = 24,\quad 6 \times 9 = 54

The ones digit of a product depends only on the ones digits of the factors.

2Test each candidate B

#6 Guess and Check 3.OA.C.7

Divide the product by each candidate: if B = 4, then 1504 / 4 = 376; if B = 9, then 1504 / 9 is not a clean three-digit number. Only B = 4 rebuilds the form A76 with tens 7 and ones 6, giving A76 = 376, so A = 3.

1504 \div 4 = 376

Dividing the product by B should rebuild the original number with the right tens and ones.

3Confirm

#6 Guess and Check 3.NBT.A.3

Check 376 x 4 by place value: 300 x 4 = 1200, 70 x 4 = 280, 6 x 4 = 24, total 1504. So A = 3 and B = 4.

376 \times 4 = 300 \times 4 = 1200, 70 \times 4 = 280, 6 \times 4 = 24 \Rightarrow 1504

Multiplying back must return the original product.

Answer: A = 3, B = 4

4 · Reviewdoes it hold up?

376 x 4 = 1504 matches exactly, the tens digit is 7 and ones digit is 6 as required, so the solution fits all conditions.

Another way: Estimate B: 1504 divided by about 376 is roughly 4, pointing straight to B = 4, then A76 = 376 gives A = 3.

Standardsmin grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Reasoning about ones digits and checking the products.
3.NBT.A.3 Multiply one-digit whole numbers by multiples of 10 — Multiplying 376 by 4 using place value to confirm.

💡Takeaway. Look at the ones digit first to narrow B, then divide to check -- classic Grade 3 detective work!

Variant 9 hard answer: A = 5, B = 3

In the multiplication below, find the digits $A$ and $B$ .

$A29 \times B = 1587$

Here $A29$ is a three-digit number whose hundreds digit is $A$ , tens digit is $2$ , and ones digit is $9$ , and $B$ is a single digit.

Show solution

1 · Understandwhat's really being asked

A three-digit number A29 (hundreds digit A, tens 2, ones 9) times a single digit B equals 1587. Find A and B.

Givens

A29 x B = 1587.
A29 has tens digit 2 and ones digit 9; A is its hundreds digit.
B is a single digit.

Unknowns

The digit A and the digit B.

Constraints

A is a digit (1 through 9, since A is the hundreds digit of a three-digit number).
B is a single digit (1 through 9).

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #6 Guess and Check

Start at the ones place: 9 times B must end in 7, which only a couple of digits do. Test each candidate B by dividing 1587 to see if it rebuilds a clean A29 form.

3 · Execute3 carry out the plan

1Use the ones digit to limit B

#2 Make a Systematic List 3.OA.C.7

The product ends in 7, and the ones digit of A29 is 9, so 9 x B must end in 7. Checking the digits, only B = 3 works for the ones place.

9 \times 3 = 27

The ones digit of a product depends only on the ones digits of the factors.

2Test each candidate B

#6 Guess and Check 3.OA.C.7

Divide the product by each candidate: if B = 3, then 1587 / 3 = 529. Only B = 3 rebuilds the form A29 with tens 2 and ones 9, giving A29 = 529, so A = 5.

1587 \div 3 = 529

Dividing the product by B should rebuild the original number with the right tens and ones.

3Confirm

#6 Guess and Check 3.NBT.A.3

Check 529 x 3 by place value: 500 x 3 = 1500, 20 x 3 = 60, 9 x 3 = 27, total 1587. So A = 5 and B = 3.

529 \times 3 = 500 \times 3 = 1500, 20 \times 3 = 60, 9 \times 3 = 27 \Rightarrow 1587

Multiplying back must return the original product.

Answer: A = 5, B = 3

4 · Reviewdoes it hold up?

529 x 3 = 1587 matches exactly, the tens digit is 2 and ones digit is 9 as required, so the solution fits all conditions.

Another way: Estimate B: 1587 divided by about 529 is roughly 3, pointing straight to B = 3, then A29 = 529 gives A = 5.

Standardsmin grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Reasoning about ones digits and checking the products.
3.NBT.A.3 Multiply one-digit whole numbers by multiples of 10 — Multiplying 529 by 3 using place value to confirm.

💡Takeaway. Look at the ones digit first to narrow B, then divide to check -- classic Grade 3 detective work!

Variant 10 hard answer: A = 2, B = 8

In the multiplication below, find the digits $A$ and $B$ .

$A61 \times B = 2088$

Here $A61$ is a three-digit number whose hundreds digit is $A$ , tens digit is $6$ , and ones digit is $1$ , and $B$ is a single digit.

Show solution

1 · Understandwhat's really being asked

A three-digit number A61 (hundreds digit A, tens 6, ones 1) times a single digit B equals 2088. Find A and B.

Givens

A61 x B = 2088.
A61 has tens digit 6 and ones digit 1; A is its hundreds digit.
B is a single digit.

Unknowns

The digit A and the digit B.

Constraints

A is a digit (1 through 9, since A is the hundreds digit of a three-digit number).
B is a single digit (1 through 9).

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #6 Guess and Check

Start at the ones place: 1 times B must end in 8, which only a couple of digits do. Test each candidate B by dividing 2088 to see if it rebuilds a clean A61 form.

3 · Execute3 carry out the plan

1Use the ones digit to limit B

#2 Make a Systematic List 3.OA.C.7

The product ends in 8, and the ones digit of A61 is 1, so 1 x B must end in 8. Checking the digits, only B = 8 works for the ones place.

1 \times 8 = 8

The ones digit of a product depends only on the ones digits of the factors.

2Test each candidate B

#6 Guess and Check 3.OA.C.7

Divide the product by each candidate: if B = 8, then 2088 / 8 = 261. Only B = 8 rebuilds the form A61 with tens 6 and ones 1, giving A61 = 261, so A = 2.

2088 \div 8 = 261

Dividing the product by B should rebuild the original number with the right tens and ones.

3Confirm

#6 Guess and Check 3.NBT.A.3

Check 261 x 8 by place value: 200 x 8 = 1600, 60 x 8 = 480, 1 x 8 = 8, total 2088. So A = 2 and B = 8.

261 \times 8 = 200 \times 8 = 1600, 60 \times 8 = 480, 1 \times 8 = 8 \Rightarrow 2088

Multiplying back must return the original product.

Answer: A = 2, B = 8

4 · Reviewdoes it hold up?

261 x 8 = 2088 matches exactly, the tens digit is 6 and ones digit is 1 as required, so the solution fits all conditions.

Another way: Estimate B: 2088 divided by about 261 is roughly 8, pointing straight to B = 8, then A61 = 261 gives A = 2.

Standardsmin grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Reasoning about ones digits and checking the products.
3.NBT.A.3 Multiply one-digit whole numbers by multiples of 10 — Multiplying 261 by 8 using place value to confirm.

💡Takeaway. Look at the ones digit first to narrow B, then divide to check -- classic Grade 3 detective work!

Variant 11 hard answer: A = 6, B = 7

In the multiplication below, find the digits $A$ and $B$ .

$A38 \times B = 4466$

Here $A38$ is a three-digit number whose hundreds digit is $A$ , tens digit is $3$ , and ones digit is $8$ , and $B$ is a single digit.

Show solution

1 · Understandwhat's really being asked

A three-digit number A38 (hundreds digit A, tens 3, ones 8) times a single digit B equals 4466. Find A and B.

Givens

A38 x B = 4466.
A38 has tens digit 3 and ones digit 8; A is its hundreds digit.
B is a single digit.

Unknowns

The digit A and the digit B.

Constraints

A is a digit (1 through 9, since A is the hundreds digit of a three-digit number).
B is a single digit (1 through 9).

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #6 Guess and Check

Start at the ones place: 8 times B must end in 6, which only a couple of digits do. Test each candidate B by dividing 4466 to see if it rebuilds a clean A38 form.

3 · Execute3 carry out the plan

1Use the ones digit to limit B

#2 Make a Systematic List 3.OA.C.7

The product ends in 6, and the ones digit of A38 is 8, so 8 x B must end in 6. Checking the digits, only B = 2 or 7 works for the ones place.

8 \times 2 = 16,\quad 8 \times 7 = 56

The ones digit of a product depends only on the ones digits of the factors.

2Test each candidate B

#6 Guess and Check 3.OA.C.7

Divide the product by each candidate: if B = 2, then 4466 / 2 is not a clean three-digit number; if B = 7, then 4466 / 7 = 638. Only B = 7 rebuilds the form A38 with tens 3 and ones 8, giving A38 = 638, so A = 6.

4466 \div 7 = 638

Dividing the product by B should rebuild the original number with the right tens and ones.

3Confirm

#6 Guess and Check 3.NBT.A.3

Check 638 x 7 by place value: 600 x 7 = 4200, 30 x 7 = 210, 8 x 7 = 56, total 4466. So A = 6 and B = 7.

638 \times 7 = 600 \times 7 = 4200, 30 \times 7 = 210, 8 \times 7 = 56 \Rightarrow 4466

Multiplying back must return the original product.

Answer: A = 6, B = 7

4 · Reviewdoes it hold up?

638 x 7 = 4466 matches exactly, the tens digit is 3 and ones digit is 8 as required, so the solution fits all conditions.

Another way: Estimate B: 4466 divided by about 638 is roughly 7, pointing straight to B = 7, then A38 = 638 gives A = 6.

Standardsmin grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Reasoning about ones digits and checking the products.
3.NBT.A.3 Multiply one-digit whole numbers by multiples of 10 — Multiplying 638 by 7 using place value to confirm.

💡Takeaway. Look at the ones digit first to narrow B, then divide to check -- classic Grade 3 detective work!

Variant 12 hard answer: A = 8, B = 8

In the multiplication below, find the digits $A$ and $B$ .

$A74 \times B = 6992$

Here $A74$ is a three-digit number whose hundreds digit is $A$ , tens digit is $7$ , and ones digit is $4$ , and $B$ is a single digit.

Show solution

1 · Understandwhat's really being asked

A three-digit number A74 (hundreds digit A, tens 7, ones 4) times a single digit B equals 6992. Find A and B.

Givens

A74 x B = 6992.
A74 has tens digit 7 and ones digit 4; A is its hundreds digit.
B is a single digit.

Unknowns

The digit A and the digit B.

Constraints

A is a digit (1 through 9, since A is the hundreds digit of a three-digit number).
B is a single digit (1 through 9).

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #6 Guess and Check

Start at the ones place: 4 times B must end in 2, which only a couple of digits do. Test each candidate B by dividing 6992 to see if it rebuilds a clean A74 form.

3 · Execute3 carry out the plan

1Use the ones digit to limit B

#2 Make a Systematic List 3.OA.C.7

The product ends in 2, and the ones digit of A74 is 4, so 4 x B must end in 2. Checking the digits, only B = 3 or 8 works for the ones place.

4 \times 3 = 12,\quad 4 \times 8 = 32

The ones digit of a product depends only on the ones digits of the factors.

2Test each candidate B

#6 Guess and Check 3.OA.C.7

Divide the product by each candidate: if B = 3, then 6992 / 3 is not a clean three-digit number; if B = 8, then 6992 / 8 = 874. Only B = 8 rebuilds the form A74 with tens 7 and ones 4, giving A74 = 874, so A = 8.

6992 \div 8 = 874

Dividing the product by B should rebuild the original number with the right tens and ones.

3Confirm

#6 Guess and Check 3.NBT.A.3

Check 874 x 8 by place value: 800 x 8 = 6400, 70 x 8 = 560, 4 x 8 = 32, total 6992. So A = 8 and B = 8.

874 \times 8 = 800 \times 8 = 6400, 70 \times 8 = 560, 4 \times 8 = 32 \Rightarrow 6992

Multiplying back must return the original product.

Answer: A = 8, B = 8

4 · Reviewdoes it hold up?

874 x 8 = 6992 matches exactly, the tens digit is 7 and ones digit is 4 as required, so the solution fits all conditions.

Another way: Estimate B: 6992 divided by about 874 is roughly 8, pointing straight to B = 8, then A74 = 874 gives A = 8.

Standardsmin grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Reasoning about ones digits and checking the products.
3.NBT.A.3 Multiply one-digit whole numbers by multiples of 10 — Multiplying 874 by 8 using place value to confirm.

💡Takeaway. Look at the ones digit first to narrow B, then divide to check -- classic Grade 3 detective work!