Question

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

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

Answer 1

Answer: (A)

$1250$

To find no. of ordered pairs $(m,n)$ such that $6^m+9^n$ is a multiple of $5.$

Sol. $:6^m+9^n;$ $m,n\in \{1,2,3,...,50\}$

${(5+1)}^m+{(10-1)}^n$

$=(5^m{+}^m{C}_1{5}^{m-1}+.....+1)+({10}^n{-}^n{C}_1{10}^{n-1}+....+(-1{)}^n)$

$=(5{\lambda }_1+1)+(10{\lambda }_2+(-1{)}^n);$ ${\lambda }_1{\lambda }_2\in I$

$=(5{\lambda }_1+10{\lambda }_2)+\{1+{(-1)}^n\}$

$=5K+\{1+{(-1)}^n\}$

The expression is divisible by $5$ only if,

$\Rightarrow 1+(-1{)}^n=0$ $\to n$ is an odd no. irrespective of the value of $m.$

$\therefore$ Total values $n$ can take $=25$

Total values $m$ can take $=50$

$\therefore$ Total no. of such ordered pairs $=25\times 50$

$=1250.$

Hence, the answer is $1250.$