## If n and m are positive integers , what is the reminder

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### If n and m are positive integers , what is the reminder

by varun289 » Sun Dec 16, 2012 3:20 am
301. If n and m are positive integers , what is the reminder when 3 ^ (4n + 2) + M is divided by 10
1. n= 2
2. m=1

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by Anindya Madhudor » Sun Dec 16, 2012 3:45 am
Let's analyze the first part of the given expression. 3^(4n + 2) can only have 9 as remainder when divided by 10. Pick any value to n to test this. The value of n is immaterial in this case.

To evaluate the whole expression, we just need to know the value of m. If we know the value of m, we can add the unit digit of m to the unit digit of 3^(4n + 2) to find the remainder.

Option I: Not sufficient
Option II: precise value of m is given. Sufficient.
Ans: B

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by aman88 » Sun Dec 16, 2012 7:49 am
IMO B

*Stmt 2 tells us that the remainder will always be 1.

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by viveksingh222 » Mon Dec 17, 2012 7:34 am
varun289 wrote:301. If n and m are positive integers , what is the reminder when 3 ^ (4n + 2) + M is divided by 10
1. n= 2
2. m=1
This question can be answered in short span of time if we know cycles of powers of 3
which are : 3,9,7,1

St I) n = 2. This makes 3^(4*2 +2 + m) = 3^(10+m). we do not know m and hence cannot figure out the unit digit.

St II) m=1 . This makes 3^(4*n +2 + 1).
4n can be 4,8,12,16...
3^(4*n +2 + 1) will be 3^7,3^11, 3^15,3^19 ..... in each case the unit digit will be 7. SUFF
Hence B

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by ceilidh.erickson » Sat Dec 22, 2012 2:05 pm
Just to elaborate on a few things that others have mentioned...

- "what is the remainder when ___ is divided by 10?" is GMAT code for "what is the units digit?" That's why we only need to worry about units digits here. It's a good idea to memorize this rule, because the GMAT uses it somewhat frequently.

- Whenever you see a units digit (UD) question involving exponents, the first thing to do is establish the pattern of the UDs. As viveksingh222 was saying, the pattern of UDs with powers of 3 is as follows:
3^1 = 3
3^2 = 9
3^3 = 7 (only worry about the UD)
3^4 = 1
3^5 = 3
3^6 = 9

So you can see that the UD repeat every 4. So any exponent that's a multiple of 4, such as 3^4n, will result in a UD of 1, so any 3^(4n + 2) will always end in a 9. Because that will always be the case for any value of n, the only unknown in this problem is the UD of m.
Ceilidh Erickson
EdM in Mind, Brain, and Education