Units Digt in Powers of 7 question

I didn't do the best job of explaining this example in class so I thought I would try to explain a similar example that will hopefully make things a bit clearer.

Question: Find the units digit of 7 raised to the 362.

Solution:

First- figure out what the question is asking.  7 raised to the 362 means 7 times itself 362 times, so you can't do that on a calculator.  The "units digit" is just asking what digit is in the ones position.

Devise a plan.  Look for patterns in powers of 7.

Carry out the plan: 

7^0 = 1, 7^1 = 7, 7^2 = 49, 7^3 = 343
7^4 = 2401, 7^5 = 16,807, 7^6 = 117,649, 7^7 = 823,543

We notice that the units digit follows a pattern: 1, 7, 9, 3, 1, 7, 9, 3,...
The patterns repeats itself in blocks of 4.

362 divided by 4 is 90.5.  360 is the closest multiple of 4 to 362 that is also smaller than 362.

So the units digit of 7^360 is 1 (because it is the first in the block of four)

The units digit of 7^361 is 7 (because it is the second in the block of 4)

The units digit of 7^362 is 9 (because it is the third in the block of 4) [and this is the final solution]

Note: there is no way to check our solution because a calculator won't give a number this big.