+864 If $m$ and $n$ are positive integers randomly chosen from the set $\{1, 2, \dots , 600\}$ with replacement, what is the probability that $2^m + 3^n$ is divisible by 11? Express your answer as a common fraction.
+1950 First, let's note that
23+31=11,23+36=737=11⋅6723+311=177155=16105⋅11
Now, from this, we can tell that
m=10a+3n=5b+1
where a and b are different numbers.
This means m has 3,13,23,...,593=60 values
And n has 1,6,11,...,596=120 values.
This means that there are 60∗120=7,200 cases that work.
In total, there are 600∗600=360,000 different cases.
Now, we just have 7200/360000=72/3600=2/100=1/50
This is a 2% chance.
Thanks! :)
