Depth · one concept from first principles · step 2 of 4
Make every prime exponent even
Find the smallest natural number A that makes × A equal to a perfect square of a natural number.
Number systemOperations
Your answer
How to solve
Strategy Organize Information in More Ways — A repeating decimal is hard to multiply, so change it to a fraction. Then write the product with letters: the denominator 9 has to cancel and the leftover factor 2 needs a partner to make an even exponent.
1STEP 1
Change the decimal to a fraction
One repeating digit after the point means divide by 9.
= 2/9
Now the product is 2/9 × A.
8.NS.A.1Organize Information In More Ways2STEP 2
Write the square condition with letters
We need 2A/9 = m². So 9 divides A; with A = 9B, 2B = m² forces m even and B = 2 × (a square).
A = 9B, 2B = m² → B = 2k² → A = 18k²
Every prime in a perfect square appears an even number of times, so the lone 2 needs another 2.
4.OA.B.4Convert To Algebrais a perfect square only when is times a perfect square.
Why?
holds no s, so the two s in can only cancel against s inside ; that forces .
🧱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.
Why?
Then , and a square holds every prime an even number of times, so needs an odd number of s and an even number of every other prime: .
Why?
is times , and since a number's bag of primes is the same however it is split, the bag for is two copies of the bag for .
🧱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.
3STEP 3
Take the smallest case and check
The smallest choice is k = 1, giving A = 18. Check the product.
2/9 × 18 = 4 = 2²
The product 4 is a perfect square, so A = 18 works.
8.NS.A.1Guess And CheckAnswer
18
Smaller candidates fail: A must be a multiple of 9, and A = 9 gives 2/9 × 9 = 2, which is not a perfect square. So 18 is the first that works.
Takeaway
Turn the repeating decimal into a fraction, then pair up every prime factor.
- Change the decimal to a fraction
- Write the square condition with letters
- Take the smallest case and check
Where next?
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