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Two rectangular prisms have the same height, but one is 38 times bigger than the other. They all have integer edge lengths and the diagonals on their faces also have integer lengths. What is the height, the length and the width of the bigger prism, knowing that it’s the smallest possible?

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  • $\begingroup$ Vf gur fznyyrfg rqtr bs gur ynetre erpgnathyne cevfz terngre guna svir uhaqerq gjragl rvtug? $\endgroup$ – Ankit Aug 21 '20 at 22:08
  • $\begingroup$ 38 times bigger is ambiguous. Do you mean in volume? Also - "They all have integer edge lengths", Do you mean they "both have integer edge lengths"? $\endgroup$ – chasly - supports Monica Aug 21 '20 at 22:55
  • $\begingroup$ @chasly-reinstateMonica You’re right. $\endgroup$ – Display maths Aug 21 '20 at 23:38
  • $\begingroup$ @Displaymaths hey can you please reply to my previous comment as well... its rot 13(Vf gur fznyyrfg rqtr bs gur ynetre erpgnathyne cevfz terngre guna svir uhaqerq gjragl rvtug?) $\endgroup$ – Ankit Aug 22 '20 at 0:17
  • $\begingroup$ @Ankit ? I’m not sure what you mean. $\endgroup$ – Display maths Aug 22 '20 at 13:20
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If I understand the question correctly, and coded correctly, then the answer is

height $1155$, width and depth are $1008, 1100$ for the smaller one and $6300, 6688$ for the larger one.

I simply

searched for every $h$, list out all the possible sizes and check whether there is one that is $38$ times larger than (which I interpret as multiply by a factor of $38$, not $39$) another.

I searched for all $h \leq 10^5$ to verify that it is indeed the smallest one.

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  • $\begingroup$ Nice, however I wanted someone to solve with number theory. $\endgroup$ – Display maths Aug 23 '20 at 1:41
  • $\begingroup$ @Displaymaths That's what the [no-computers] tag is for. $\endgroup$ – Paul Panzer Aug 23 '20 at 2:03

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