Quandaries
and Queries
Name: Alex
Who is asking: Student
Level of the question: Secondary
Question: What is the unit's digit of the following problem?
3155 + 7412 (3 to the 155th power + 7 to the 412th power.)
Hi Alex,
I can get you started by considering 3155. First look at some powers of 3 and observe the units digit.
| power of 3 | units digit |
|---|---|
| 31 | 3 |
| 32 | 9 |
| 33 | 7 |
| 34 | 1 |
| 35 | 3 |
| 36 | 9 |
| 37 | 7 |
| 38 | 1 |
| 39 | 3 |
| … | … |
The point to notice here is that the units digit cycles in a cycle of length four, 3,9,7,1,3,9,7,1, … If you continue the table above for 155 rows then, since 155/4 = 38 with a remainder of 3, will see 38 copies of the cycle 3,9,7,1 and then three more rows where the units digits are 3,9 and 7. Thus the units digit of 3155 is 7.
Penny
