Is the following statement true or false?: There are integers $x$, $y$ and $z$ such that $15$ divides $2^x \cdot 3^y \cdot 7^z$

Question

My son asked me for help on this question but I have forgotten how to do it. Could you please save me from looking like an idiot in front of my son, it would be very much appreciated. Thank you

Answer 1

The statement is false. Consider the prime factorization of $15$ is $5 \cdot 3$, but we have $5 \cdot 3\cdot k=(2^x)(3^y)(7^z)$ The LHS has a factor of $5$, but the RHS does not. Therefore, since 5 is prime, we have that there are no solutions.