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Source: Indian TST 2019 Practice Test 2 P2

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643 posts

#1 h Jul 17, 2019, 12:57 PM • 1 Y

Y by Adventure10

Determine all positive integers $m$ satisfying the condition that there exists a unique positive integer $n$ such that there exists a rectangle which can be decomposed into $n$ congruent squares and can also be decomposed into $m+n$ congruent squares.

This post has been edited 1 time. Last edited by ayan.nmath, Jul 17, 2019, 1:03 PM

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#2 h Nov 18, 2019, 7:30 PM • 1 Y

Y by Adventure10

We claim that all such $m$ are $p$ , $2p$ , $4p$ , where $p$ is an odd prime. Note that the problem is equvivalent to the following: find all $m$ , for wish there exist exactly one positive divisor $d$ of $m$ , which can be expressed as $a^2-b^2$ , where $(a,b)=1$ . Note that if $m$ has $2$ or more odd divisors, then it can not satisfy the condition:every positive integer $2k+1=(k+1)^2-k^2$ and $(k+1)$ and $k$ are clearly coprime. So we only need to consider numbers $m=2^kp$ , where p is a prime. But $8=3^2-1$ , so $k\leq 2$ . It's easy to verify that $p$ , $2p$ , $4p$ satisfy the condition.

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AN1729

#3 h Jul 22, 2023, 3:17 PM

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Good Problem!

This post has been edited 2 times. Last edited by AN1729, Oct 23, 2024, 12:24 PM

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