Problem:
Two distinct numbers a and b are chosen randomly from the set {2,22,23,β¦,225}. What is the probability that logaβb is an integer?
Answer Choices:
A. 252β
B. 30031β
C. 10013β
D. 507β
E. 21β
Solution:
Let a=2j and b=2k. Then
logaβb=log2jβ2k=log2jlog2kβ=jlog2klog2β=jkβ
so logaβb 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 βj25ββ. Because jξ =k, the number of possible values of k for each j is βj25βββ1. Hence the total number of ordered pairs (a,b) is
j=1β25β(βj25βββ1)=24+11+7+5+4+3+2(2)+4(1)=62
Since the total number of possibilities for a and b is 25β
24, the probability that logaβb is an integer is
25β
2462β=30031ββ
The problems on this page are the property of the MAA's American Mathematics Competitions