What is the smallest natural number by which $90$ should be multiplied to make it a perfect square?

What is the smallest natural number by which $90$ should be multiplied to make it a perfect square?
  1. $10$
  2. $5$
  3. $2$
  4. $90$

Solution

$90 = 2 \times 3^{2} \times 5$. The unpaired primes are $2$ and $5$, so multiply by $2 \times 5 = 10$ to get $900 = 30^{2}$.

Asked in: IMO

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