<s>[INST] Finding Smallest x and y to Satisfy Equation Find the smallest natural numbers $x$ and $y$ such that $$7^2x=5^3y$$
I'm unsure how to proceed with this question. Could someone explain the process for determining the answer?
Added from the comments:
$x=5^3$ would satisfy $5^3$ divides $7^2x$
 [/INST] First consider left side. $5^3$ divides right side, so $5^3$ must also divides left side. Now $7^2$ and $5^3$ are relatively prime, so $5^3$ occurs in $x$, in other word:
$$x=5^3 \cdot x'$$
For some natural $x'$. $x'$ is natural, so $x' \geq 1$, so:
$$x=5^3 \cdot x' \geq 5^3$$
The same way we get:
$$y \geq 7^2$$
But if we put $x=5^3$ and $y=7^2$ the equation is satisfied, so $x,y$ are the smallest possible solutions.
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