<s>[INST] Probability that $7^m+7^n$ is divisible by $5$ 
If $m,n$ are chosen from the first hundred natural numbers with replacement, the probability that $7^m+7^n$ is divisible by $5$ is?

$$7^m+7^n=7^m(1+7^{n-m}), n\ge m$$
The above expression is divisible by $5$ only if $n-m=4k+2$. The max value of $k$ is $24$.
So is the number of possibilities $48$? ($24$ for each $n>m$ and $m>n$) 
 [/INST] The possibilities of ending digits of $7^m$ or $7^n$ : $9,3,1,7$
Favourable cases : $(9,1),(3,7),(1,9),(7,3)$
Probability : $\frac{4}{4\times 4} =\frac{1}{4}$
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