I've got this GRE math question: The integer y is positive. If $6^y$ is a factor of $(2^{14})(3^{24})$, then what is the greatest possible value of y?
The answer is 14. Why? I've been away from math too long so the answer doesn't compute in my brain. Thanks.
Our expression $2^{14}\cdot 3^{24}$ can be rewritten as: $$2\cdot 3\cdot 2^{13}\cdot 3^{23}$$ $$6\cdot 2^{13}\cdot 3^{23}$$ To make another $6$ factor, we just borrow a $2$ and a $3$ like this: $$6^2\cdot 2^{12}\cdot 3^{22}$$ How many $6$s can we make? We can make as many $6$s as we want until either $2$ or $3$ runs out. In this case we can make as many $6$s as we want until $2$ runs out. $$6^{14}\cdot 3^{10}$$ Therefore $y=14$