UNITS DIGIT

shrey2287 wrote:What is the units digit of the solution to 177^28 - 133^23 ?

(A) 1
(B) 3
(C) 4
(D) 6
(E) 9

177^28 ends in 1 (units digit of powers of 7 follow the cycle 7, 9, 3, 1)
and 133^23 ends in 7 (units digit of powers of 3 follow the cycle 3,9,7,1)

So, a number ending in 1 - a number ending in 7 ends in 4.

The correct answer is 4, (C).

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