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roger federer
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Let's start by analyzing the problem:roger federer wrote:If n is a positive integer, what is the remainder when 3^(8n+3) + 2 is divided by 5?
A. 0
B. 1
C. 2
D. 3
E. 4
we've got some gigantic number and we're asked to find the remainder when we divide by 5. On the GMAT, will we ever have to calculate gigantic numbers? NO! So, there must be another way to solve.
One key thing to note is that we're dividing by 5; this makes us very happy! Why? Because to find the remainder when you divide by 5, you only need the last digit of the number.
So, we think: to solve the question, we just need to figure out the last digit of the gigantic number.
We then see that we're dealing with powers of 3. Let's jot some down on our scrap paper and look for a pattern:
3^1 = 3
3^2 = 9
3^3 = 27
3^4 = 81
3^5 = ...3
3^6 = ...9
at this point we see that we have a 4 step repeating pattern: the units digit will cycle through 3, 9, 7 and 1.
Now that we have our pattern, let's apply it to the question at hand:
3^(8n+3) + 2
For simplicity, let's let n=1. That gives us:
3^11 + 2
Working through our repeating pattern, we see that 11 is going to be the 3rd step and that the units digit of 3^11 is 7.
Accordingly, the units digit of 3^11 + 2 will be 7 + 2 = 9.
Finally, when we divide 9 by 5, we have a remainder of 4... choose (E).
