Question

The number of ordered pairs $(m,n)m,n\in 1,2,...........50$such that $6^n+9^m$ is a multiple of $5$

• A. $2500$
• B. $1250$
• C. $625$
• D. $500$

Answer 1

The correct option is B $1250$

The key is to find a relation between $m$ and $n$. For this question, if we take modulo $5$,we get $1^m+(-1{)}^n=1+(-1{)}^n$ The first thing to notice is that this is independent of $m$. Therefore, every value of $m$ is possible. Thus, there are $50$ possible $m$ values. To make this equal to $0$, we want $n$ to be odd. Thus, there are $25$ possible $n$ values. Therefore, there are $\left(50\right)\left(25\right)=1250$ number of possible pairs.