Find the smallest whole number..

Question

This question is related to prime factorisation.

How do I find the smallest whole number $b$ for which $240/b$ is a factor of $252$?

\begin{align*} 240 &= 2^4 \times 3 \times 5 \\ 252 &= 2^2 \times 3^2 \times 7 \\ \end{align*}

Does the question mean the result from the division is a factor of 252? I'm a bit confused..

Answer 1

find $gcd(240,252)$. it is equal to $12$. so $b = 240/12 = 20$

Answer 2

In answering this question we may query the largest factor of $240$ that is also a factor of $252$. Clearly, this is the GCD of the two numbers, which works out to be $2^2\cdot 3=12$ (using the prime factorisations given).

Since $12$ is the largest factor of $240$ that is also a factor of $252$, $240/12=20$ must be the smallest $b$ such that $240/b$ is a factor of $252$.