Problem:
Two distinct numbers a and b are chosen randomly from the set {2,22,23,…,225}. What is the probability that logab is an integer?
Answer Choices:
A. 252
B. 30031
C. 10013
D. 507
E. 21
Solution:
Let a=2j and b=2k. Then
logab=log2j2k=log2jlog2k=jlog2klog2=jk
so logab is an integer if and only if k is an integer multiple of j. For each j, the number of integer multiples of j that are at most 25 is ⌊j25⌋. Because j=k, the number of possible values of k for each j is ⌊j25⌋−1. Hence the total number of ordered pairs (a,b) is
j=1∑25(⌊j25⌋−1)=24+11+7+5+4+3+2(2)+4(1)=62
Since the total number of possibilities for a and b is 25⋅24, the probability that logab is an integer is
25⋅2462=30031
The problems on this page are the property of the MAA's American Mathematics Competitions