Problem: If 2137753 is multiplied out, the units' digit in the final product is:
Answer Choices:
A. 1
B. 3
C. 5
D. 7
E. 9
Solution:
The final digits for (2137)n are respectively 1 if n=0,7 if n=1,9 if n=2, and 3 if n=3. For larger values of n these digits repeat in cycles of four. ∴(2137)753=(2137)4⋅188+1=(21374)188⋅(2137)1. The final digit of (21374)188 is 1 and the final digit of (2137) 1 is 7. The product yields a final digit of 1⋅7=7.