Problem Solving

[GMAT math practice question]

f(x) denotes the remainder when x is divided by 10. For example, f(5) = 5, f(92) = 2 and f(271) = 1. What is the value f(2^2006) + f(3^2006) + f(5^2006) + f(7^2006) + f(9^2006)?

A. 24
B. 26
C. 28
D. 30
E. 32

=>
When we repeat multiplying a number every four times, we have the same unit digit.
2^{2006} has the same unit digit of 2^2, which is 4, since 2006 = 4·501 + 2.
3^{2006} has the same unit digit of 3^2, which is 9, since 2006 = 4·501 + 2.
5^{2006} has the same unit digit of 5^2, which is 5, since 2006 = 4·501 + 2.
7^{2006} has the same unit digit of 7^2, which is 9, since 2006 = 4·501 + 2.
9^{2006} has the same unit digit of 9^2, which is 1, since 2006 = 4·501 + 2.

f(2^{2006})+ f(3^{2006})+ f(5^{2006})+ f(7^{2006})+ f(9^{2006}) = 4 + 9 + 5 + 9 + 1 = 28.

Therefore, C is the answer.
Answer: C

This can be solved by Cyclicity of remainders...

Cyclicity of 2 IS 4 : 2, 4, 8, 6
Cyclicity of 5 IS 1: 5
Cyclicity of 3 IS 4: 3, 9, 7, 1
Cyclicity of 7 IS 4 : 7, 9, 3, 1
Cyclicity of 9 IS 2 : 9, 1

Divide 2006 by the count of cyclicity.. and find remainders...

f(2^{2006})+ f(3^{2006})+ f(5^{2006})+ f(7^{2006})+ f(9^{2006})
= 4 + 9 + 5 + 9 + 1
= 28

Hence C