Problem:
How many positive two-digit integers are factors of 224−1 ?
Answer Choices:
A. 4
B. 8
C. 10
D. 12
E. 14
Solution:
Factoring results in the following product of primes:
224−1=(212−1)(212+1)=(26−1)(26+1)(24+1)(28−24+1)=63⋅65⋅17⋅241=3⋅3⋅5⋅7⋅13⋅17⋅241
The two-digit integers that can be formed from these prime factors are:
17,3⋅17=51,5⋅17=8513,3⋅13=39,5⋅13=65,7⋅13=913⋅7=21,5⋅7=35,3⋅3⋅7=633⋅5=15, and 3⋅3⋅5=45
Thus there are 12 positive two-digit factors.
The problems on this page are the property of the MAA's American Mathematics Competitions