alanforde800Maximus wrote:What is the units digit of 9^19 - 7^15?

a) 2
b) 4
c) 5
d) 6
e) 7

Please assist with above problem.

If you're not sure how to proceed with this question, you can eliminate 2 answer choices in a matter of seconds by recognizing two things:
9^19 = some ODD integer, since 9^19 is a product consisted entirely of odd integers
Likewise, 7^15 = some ODD integer, since 7^15 is a product consisted entirely of odd integers

So, 9^19 - 7^15 = (some ODD integer) - (some ODD integer) = EVEN

So, the correct answer is EVEN.
Eliminate C and E, and make a guess.

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Cheers,
Brent

alanforde800Maximus wrote:What is the units digit of 9^19 - 7^15?

a) 2
b) 4
c) 5
d) 6
e) 7

Please assist with above problem.

We need to separately determine the units digit of 9^19 and 7^15.

Since all odd powers of 9 have a units digit of 9, and all even powers of 9 have a units digit of 1, the units digit of 9^19 is 9.

To determine the units digit of 7^15, we look at the pattern of the units digits of 7^n for positive integer values of n. That is, let's look at the pattern of the units digits of powers of 7.

7^1 = 7

7^2 = 9

7^3 = 3

7^4 = 1

7^5 = 7

The pattern of any base of 7 repeats every 4 exponents. The pattern is 7-9-3-1. In this pattern, all positive exponents that are multiples of 4 will produce a 1 as its units digit. The closest multiple of 4 to 15 is 16. Thus, 7^16 will have a 1 as its units digit and, thus, 7^15 will have a 3 as its units digit.

Thus, the units digit of 9^19 - 7^15 = 9 - 3 = 6

Answer: D