Problem: Let m and n be any two odd numbers, with n less than m. The largest integer which divides all possible numbers of the form m2−n2 is:
Answer Choices:
A. 2
B. 4
C. 6
D. 8
E. 16
Solution:
m=2r+1 where r=0,±1,±2,…
n=2 s+1 where s=0,±1,±2,…
m2−n2=4r2+4r+1−4 s2−4 s−1
=4(r−s)(r+s+1), a number certainly divisible by 4.
If r and s are both even or both odd, r−s is divisible by 2 .
If r and s are one even and one odd, then r+s−1 is divisible by 2 .
Thus m2−n2 is divisible by 4⋅2=8