shrey2287 wrote: ↑Tue Sep 20, 2011 6:44 pm
What is the units digit of the solution to 177^28 - 133^23 ?
(A) 1
(B) 3
(C) 4
(D) 6
(E) 9
Solution:
To solve this problem, we need to use the following two facts:
1) If u is the units digit of a positive integer n, then n^m and u^m have the same units digit.
2) Let m and n be the units digits of M and N. If M - N is positive but m - n is negative, then the units digit of M - N is m - n + 10.
Using the first fact, we see that 177^28 has the same units digit as 7^28, and 133^23 has the same units digit as 3^23.
Since the units digit pattern of powers of 7 is 7-9-3-1,we see that 7^28 has units digit of 1. Likewise, since the units digit pattern of powers of 3 is 3-9-7-1, then 3^23 has a units digit of 7.
Using the second fact, we see that the units digit of 177^28 - 133^23 is 1 - 7 + 10 = 4.
Answer: C
