Problem:
Given that 38⋅52=ab, where both a and b are positive integers, find the smallest possible value for a+b.
Answer Choices:
A. 25
B. 34
C. 351
D. 407
E. 900
Solution:
Since a must be divisible by 5, and 38⋅52 is divisible by 52, but not by 53, we have b≤2. If b=1, then
ab=(3852)1=(164,025)1 and a+b=164,026
If b=2, then
ab=(345)2=4052 so a+b=407,
which is the smallest value.
Answer: D.
The problems on this page are the property of the MAA's American Mathematics Competitions