Problem:
Find the units digit in the decimal expansion of (15+220)19+(15+220)82.
Answer Choices:
A. 0
B. 2
C. 5
D. 9
E. none of these
Solution:
Let d1=a+b and d2=a−b, where a=15 and b=220. Then using the binomial theorem, we may obtain
d1n+d2n=2[an+(2n)an−2b+(4n)an−4b2+…]
where n is any positive integer. Since fractional powers of b have been eliminated in this way, and since a and b are both divisible by 5, we may conclude that d1n+d2n is divisible by 10.
We now apply the above result twice, taking n=19 and n=82. In this way we obtain
d119+d219=10k1 and d182+d282=10k2,
where k1 and k2 are positive integers. Adding and rearranging these results gives
d119+d182=10k−(d219+d282),
where k=k1+k2. But d2=15−220=15+2205<31.
Therefore, d219+d282<1. It follows that the units digit of 10k−(d219+d282) is 9.