Question

The number of ordered pairs $(m,n)$, where $m,n\in \{1,2,3,\dots ,50\}$, such that $6^m+9^n$ is a multiple of $5$ is

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

Answer 1

Answer: (A)

$1250$

For a number to be divisible by $5$ the last digit shall be a multiple of $5$
Any power of 6 has its last digit(or unit's digit) equal to 6.
The power of $9$ of the form
$4p+1$ has it's last digit as 9
$4p+2$ has it's last digit as 1
$4p+3$ has it's last digit as 9
$4p$ has it's last digit as 1
Thus $6^m+9^n$ will be a multiple of $5$ only if $m$ is any number from the given set and $n$ is a number of the form $4p+1$ or $4p+3$.
Therefore the number of ways to select $m$ is $(\frac{50}{1})$ and the number of ways to select $n$ from the given set is $(\frac{25}{1})$
hence the total number of ordered pairs $(m,n)$ is $(\frac{50}{1})\times (\frac{25}{1})=50\times 25=1250$