My son is having this problem:
Smallest positive integer that is divisible by $2$, $3$ and $5$, and that is also a perfect square and perfect cube.
He was busy using Excel, but I thought: Isn't there a faster way?
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$\begingroup$ $$(2\cdot3\cdot5)^{6m}$$ $\endgroup$– lab bhattacharjeeCommented Nov 20, 2018 at 12:13
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$\begingroup$ @labbhattacharjee: Since he said "smallest", my answer would be $(2\cdot 3\cdot 5)^6$, $\endgroup$– GEdgarCommented Nov 20, 2018 at 12:16
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2 Answers
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Since number has to be divisible by $2,3,5$ therefore it should be divisible by $30$. Since you want a perfect square and a perfect cube therefore your number should be of the form $30^{6t}$ . Where t is the integral number satisfying all such conditions. In your case $t=1$ , therefore answer is $30^6$.
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$\begingroup$ It took me a long time to edit 😥, I could have posted it before lab bhattacharjee. $\endgroup$ Commented Nov 20, 2018 at 12:21
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$\begingroup$ Thanks for upvote , next time I will try my best not to delay. $\endgroup$ Commented Nov 20, 2018 at 12:38
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If $n$ is an integer, then
- $n^2$ is a perfect square
- $n^3$ is a perfect cube
An integer that has both these properties is $n^{2\cdot 3}=n^{6}$, since $$n^{6}=\underset{\color{red}{\text{square}}}{\underbrace{(n^{3})^\color{red}{2}}}=\underset{\color{green}{\text{cube}}}{\underbrace{(n^{2})^\color{green}{3}}}$$ You also want this to be divisible by $2,3,5$, these are prime numbers so the smallest non-zero such number is $$(2\cdot 3\cdot 5)^6=30^6=729 000 000$$
answered Nov 20, 2018 at 12:23