 Quandaries and Queries

I am a student
secondary

What is the ones digit of 3 to the 79 power?  (hint: compute the first 12 powers of 3 and look for a pattern in the ones digits.)  I have been trying for over 2 hours with this problem, I know it is probably a simple solution but for some reason I tend to over analyze and make it harder than it should be.  If you can help I would be appreciative.

Candice

Hi Candice,

I'm going to compute powers of 3 and record them in a table but only record the ones digit.

power of 3 ones digit
1 3
2 9
3 7
4 1
5 3
6 9
7 7
8 1
9 3

What I see is that the ones digits are repeating in a pattern, 397139713971···. The pattern is 3971 and hence the pattern length is 4 digits.

So what would the table above look like if I extended it to 79 rows?

79 divided by 4 is 19 with a remainder of 3 so the pattern would repeat 19 times and then there would be 3 more rows. The ones digits in these three rows are 3, 9 and 7 so the ones digit of 3 to the 79 is 7.

power of 3 ones digit
1 3
2 9
3 7
4 1
. .
. .
73 3
74 9
75 7
76 1
77 3
78 9
79 7

Penny   