Prove that a perfect square must end in one of the following pairs of digits: 00, 01, 04, 09, 16, 21, 24, 25, 29, 36, 41, 44, 49, 56, 61, 64, 69, 76, 81, 84, 89, 96.
[Hint: Because x2 ≡ (50 + x)2 (mod 100) and x2 ≡ (50 − x)2 (mod 100), it suffices to examine the final digits of x2 for the 26 values x = 0, 1, 2,…, 25.]
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