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

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) None
Answer 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
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