To determine the remainder when (65^{2016} + 67^{2016}) is divided by a certain modulus, we can break down the problem as follows:
-
Compute (65^{2016} mod 100):
- (65 equiv 5 mod 100).
- We need to find (5^{2016} mod 100).
- (65 equiv 3 mod 4).
- Since (phi(100) = 40), we need to consider (5) modulo 4 and 25.
- (65 equiv 25 mod 100).
- Using Chinese Remainder Theorem, we find (65^{2016} equiv 25 mod 100).
-
Compute (67^{2016} mod 100):
- (67 equiv -3 mod 100).
- We need to find (-3^{2016} mod 100).
- (67 equiv 17 mod 25).
- Using Euler’s theorem, we find (17^{2016} equiv 6 mod 25).
- Using the Chinese Remainder Theorem, we find (67^{2016} equiv 81 mod 100).
- Sum the results:
- (65^{2016} equiv 25 mod 100)
- (67^{2016} equiv 81 mod 100)
- Sum: (25 + 81 = 106)
- (106 mod 100 = 6)
Thus, the remainder when (65^{2016} + 67^{2016}) is divided by 100 is (boxed{6}).
