A and B are two positive numbers that end in 9. Prove that A2-B2 is divisible by 40.
Let A = 10N-9 and B = 10M-9
Then,
A^2-B^2
= 100 N^2 -180N + 100N^2 - 180 M
= 20N(5N-9) + 20M(5M -9)
If N is even, N(5N-9) is even
If N is odd, N(5N-9) is even
Therefore, N(5N-9) is always even.
Likewise, M(5M-9) is always even.
Consequently, A^2-B^2 is always divisible by 20*2=40
Edited on June 21, 2022, 5:11 am
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