-
Two distinct numbers a and b are chosen randomly from the set
{2, 22 , 23 , . . . , 225 }. What is the probability that loga b is an integer?
(A)
2
25
(B)
31
300
(C)
13
100
(D)
7
50
(E)
1
2
2005 AMC 12 A, Problem #23— “Let a = 2j and b = 2k . . .”
- Solution (B) Let a = 2j and b = 2k . Then
loga b = log2j 2k =
log 2k
k log 2
k
=
= ,
j
log 2
j log 2
j
so loga b is an integer if andj only
k if k is an integer multiple of j. For each j, the number of integer jmultiples
k
25
of j that are at most 25 is 25
.
Because
j
=
6
k,
the
number
of
possible
values
of
k
for
each
j
is
− 1.
j
j
Hence the total number of ordered pairs (a, b) is
º
25 µ¹
X
25
j=1
j
¶
−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 loga b is an integer is
31
62
=
.
25 · 24
300
Difficulty: Hard
NCTM Standard: Data Analysis and Probability Standard: Develop and evaluate inferences and predictions that are based on
data
Mathworld.com Classification:
Probability and Statistics > Probability > Probability
63