# SOLUTION: Find the remainder when the sum 4^37 + 6^37 is divided by 25. A) 5 B) 10 C) 15 D) 20 E) None

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 Question 360749: Find the remainder when the sum 4^37 + 6^37 is divided by 25. A) 5 B) 10 C) 15 D) 20 E) NoneAnswer by Sphinx pinastri(17)   (Show Source): You can put this solution on YOUR website!First, notice that 6 = 10 - 4 and rewrite the problem as: (4^37 + (10 - 4)^37) modulo 25 Expand the last term using binomial formula: (4^37 + 10^37 + ... + 37 x 10 x 4^36 - 4^37) modulo 25 Notice, that 1st and last terms cancel out and all terms with 10^2 and higher power are equal 0 modulo 25. So we have: (37 x 10 x 4^36) modulo 25 Let's look at 4^36. 4^36 = 2^72 = 4 x 1024^7 1024^7 = (1025 - 1)^7 (1025 - 1)^7 modulo 25 = -1 modulo 25 So the problem is reduced to: (-37 x 4 x 10) modulo 25 = (50 - 37) modulo 25 x 40 modulo 25 = 13 x 15 modulo 25 = 195 modulo 25 = 20