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Let $a,b \in \mathbb{N}$ such that $2a=3b$. Show that $2|b$ and $3|a$.

My Approach:

My approach to this question is to find an expression for $b$ in a way that the expression is divisible by $2$. Then, find an expression for $a$ in such a way that the expression is divisible by $3$. I tried several calculations but none of them seems to go anywhere.

amundi12
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  • There's a theorem: If $p$ is a prime, and $p\mid ab$, then either $p\mid a$ or $p\mid b$ (or both). (Example: $5\mid10\times17$. Thus, either $5\mid10$ or $5\mid17$.) – Akiva Weinberger Dec 11 '14 at 03:57

1 Answers1

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Since 2 divides 3b, either 2 divides 3 or 2 divides b. Since 2 doesn't divide 3, 2 must divide b.

Suzu Hirose
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