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Problem 295

Number theory Difficulty 2.4 Multiple choice CEMC Fermat · Canada · 2022

If $\$2^{11}\times 6^5=4^x\times$
3^y for some positive integers x $and$ y $, then the value of$ x+y$ is

Pick one

Official solution

Manipulating the left side, $\$2^{11} \times$ 6^5 = $2^{11} \times$ (2 $\times$ 3)^5 = $2^{11} \times$ 2^5 $\times$ 3^5
= $2^{16} \times$ 3^5$.

Since $4^x $\times$ 3^y = $2^{16} \times$
3^5 $and$ x $and$ y $are positive integers, then$ y = 5 $(because$ 4^x$ has no factors of 3).

This also means that $4^x =
2^{16}$.

Since $4^x = (2^2)^x = 2^{2x}$ , then
$4^x = 2^{16}$ gives $2^{2x} = 2^{16}$ and so $2x = 16$ or $x=8$ .

Therefore, $x + y = 8 + 5 = 13$ .

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.