Problem: If p is a positive integer, then 2p−53p+25 can be a positive integer, if and only if p is:
Answer Choices:
A. at least 3
B. at least 3 and no more than 35
C. no more than 35
D. equal to 35
E. equal to 3 or 35
Solution:
Let (3p+25)/(2p−5)=n, a positive integer. Then, 3p+25=kn, 2p−5=k;∴k(2n−3)=65=1⋅65=5⋅13.
∴k=1,65,5, or 13 and 2n−3=65,1,13, or 5 correspondingly.
∴2p=5+1,5+65,5+5, or 5+13;∴p=3,35,5, or 9.
∴ choice (B) is correct.