From Kaushik Srinivasan Find all n that 3n + 7n is divisible by 10 We notice that positive powers of 3 end in 3, 9, 7, 1 (i.e 31, 32, 33, 34)
and these digits repeat themselves. Also the powers of 7 end in 7, 9, 3, 1 repectively. Hence when the odd powers of 3 and 7 are added we get a zero
as the last digit thus divisible by 10. Hence when n is odd 3n + 7n is divisible
by 10.
Part solutions were received from a number of our members. Ngoc
Tran, Clement Goh (River Valley High School) and Andrei Lazanu
(School 205 Bucharest). You were all able to spot patterns but no
one explained why or how you might predict what will happen. For
example why does 3 4n always end in 1?
A spreadsheet was a really useful tool for looking for
patterns by quickly calculating powers of each of the numbers and
then looking for patterns, but not for explaining why. Why is the
power always of the form 2k+1 of 4k+2 except for 5 n +
5 n ?
Here are some combinations (n and k are integers):
| 1 n + 1 n |
never (why?) |
| 1 n + 2 n |
never (why?) |
| 1 n + 3 n |
when n = 4k+2 (why?) |
| 1 n + 4 n |
never |
| 1 n + 5 n |
ever |
| 1 n + 6 n |
never |
| 1 n + 7 n |
when n = 4k+2 |
| 1 n + 8 n |
never |
| 1 n + 9 n |
when n = 2k+1 |
| 2 n + 2 n |
never |
| 2 n + 3 n |
never |
| 2 n + 4 n |
when n = 4k+2 |
| 2 n + 5 n |
never |
| 2 n + 6 n |
when n = 4k+2 |
| 2 n + 7 n |
never |
| 2 n + 8 n |
when n = 2k+1 |
| 2 n + 9 n |
never |
| 3 n + 3 n |
never |
| 3 n + 4 n |
never |
| 3 n + 5 n |
never |
| 3 n + 6 n |
never |
| 3 n + 7 n |
when n = 2k+1 |
| 3 n + 8 n |
never |
| 3 n + 9 n |
when n = 4k+2 |
| 4 n + 4 n |
never |
| 4 n + 5 n |
never |
| 4 n + 6 n |
when n = 2k+1 |
| 4 n + 7 n |
never |
| 4 n + 8 n |
when n = 4k+2 |
| 4 n + 9 n |
never |
| 5 n + 5 n |
when n=k |
| 5 n + 6 n |
never |
| 5 n + 7 n |
never |
| 5 n + 8 n |
never |
| 5 n + 9 n |
never |
| 6 n + 6 n |
never |
| 6 n + 7 n |
never |
| 6 n + 8 n |
when n = 4k+2 |
| 6 n + 9 n |
never |
| 7 n + 7 n |
never |
| 7 n + 8 n |
never |
| 7 n + 9 n |
never |
| 7 n + 9 n |
when n = 4k+2 |
| 8 n + 8 n |
never |
| 8 n + 9 n |
never |