Depth · one concept from first principles · step 2 of 4
Smallest square that is a common multiple
Natural numbers , , and satisfy . Find the smallest possible value of .
Number systemExpressions
Your answer
How to solve
Strategy Organize Information in More Ways — Rewrite 360 and 1500 as products of prime powers. Then c² must contain the least common multiple, and on top of that every exponent must be even. Raise each odd exponent by one and take the square root.
1STEP 1
Factor both numbers into primes
Break each coefficient into prime powers.
360 = 2³ × 3² × 5, 1500 = 2² × 3 × 5³
Now the exponents can be compared prime by prime.
6.EE.A.1Organize Information In More Ways2STEP 2
c² is a common multiple
Since c² = 360a = 1500b, c² is a common multiple, so a multiple of the LCM: take the larger exponent of each prime.
2³ × 3² × 5³ = 9000
c² is a multiple of 9000.
6.NS.B.4Identify Subproblems3STEP 3
Make every exponent even
A square has even exponents. Raise the odd exponents 3 on 2 and on 5 up to 4; the exponent 2 on 3 is already even.
c² = 2⁴ × 3² × 5⁴ = 90000
This is the smallest square that is a multiple of 9000.
6.EE.A.1Organize Information In More WaysThe smallest perfect square that is a multiple of is .
Why?
is a multiple of both and , so it sits on both lists of multiples, which means it is a multiple of their least common multiple .
🧱Common multiples are the multiples of the least oneThe numbers that appear on two multiple lists at once are exactly the multiples of the first number where the lists agree, so nothing between them can be shared.
Why?
Squaring is , which uses every prime of twice, so each exponent in is a count added to itself and is even.
🧱An exponent counts repeated multiplicationAn exponent records only how many times a factor is used, so multiplying two powers of the same base just adds those counts together.
Why?
A multiple of must contain at least three s, two s and three s, and since its prime breakdown is unique, the smallest even counts that reach these are , and .
🧱Every number has one prime factorizationKeep splitting a number into factors and you always land on the same bag of primes, no matter which split you tried first.
4STEP 4
Take the square root
Halve each exponent.
c = 2² × 3 × 5² = 300
The smallest c is 300.
6.EE.A.1Identify SubproblemsAnswer
300
300² = 90000, and 90000 ÷ 360 = 250 and 90000 ÷ 1500 = 60 are both natural numbers, so a = 250, b = 60 work.
Takeaway
For a square, look at the exponents: every one must be even.
- Factor both numbers into primes
- c² is a common multiple
- Make every exponent even
- Take the square root
Where next?
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