If n and m are positive integers what is the remainder when 3^(4n +2) + m is divided by 10?
1) n = 2
2) m =1
Official answer is B, 2) along is sufficient but I don't understand why. Any takers?
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sk818020
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First you'll notice a pattern;
3^1=3
3^2=9
3^3=27
3^4=81
3^5=243
3^6=729
3^7=2187
3^8=6561
3^10=59049
Some other thing to note;
3^(4n+2)
- 3 will always in this example be raised to a positive power. So you should tell yourself to look for patterns in the even powers.
- The power that 3 will be raised to will always be 2 or a multiple of 4, plus two. So, 2, 6=[4(1)+2], 10=[4(2)+2] and so forth. Note this and then look at the numbers above. 2, 6, and 10 all end in 9. When you notice a pattern like this you need to take advantage of it.
Going off the notion that 3^(4n+2) will have a nine in it's digits place we only need to know what the value of m is to solve the question.
(2) tells us what m is so we can solve the problem. Thus, 2 is sufficient.
3^1=3
3^2=9
3^3=27
3^4=81
3^5=243
3^6=729
3^7=2187
3^8=6561
3^10=59049
Some other thing to note;
3^(4n+2)
- 3 will always in this example be raised to a positive power. So you should tell yourself to look for patterns in the even powers.
- The power that 3 will be raised to will always be 2 or a multiple of 4, plus two. So, 2, 6=[4(1)+2], 10=[4(2)+2] and so forth. Note this and then look at the numbers above. 2, 6, and 10 all end in 9. When you notice a pattern like this you need to take advantage of it.
Going off the notion that 3^(4n+2) will have a nine in it's digits place we only need to know what the value of m is to solve the question.
(2) tells us what m is so we can solve the problem. Thus, 2 is sufficient.
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Patrick_GMATFix
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Hi Skalevar,skalevar wrote:If n and m are positive integers what is the remainder when 3^(4n +2) + m is divided by 10?
1) n = 2
2) m =1
The key is to solving this Q quickly is to come up with a strong rephrase. Recognize 2 general rules:
1) remainder when an integer is divided by 10 is the units digit of that integer. So the question can be rephrased as: What is the units digit of 3^(4n+2) + m?
2) units digit of an exponential expression follows a cyclical pattern as the exponent is incremented
In this case rephrasing is key. Since 3^(4n+2) is 9^(2n+1). In other words, it is 9 raised to an odd exponent. Regardless of the exponent, the units digit of 9^(odd) is 1. So to determine the units digit of 3^(4n+2) + m, we only need the units digit of m.
Rephrase: What is the units digit of m?
Statement 2 gives us this info so it's sufficient. The answer is B.
A more detailed explanation and a step-by-step video solution is available at GMATPrep Question 1395. If you struggle with advanced remainder problems, set topic='Number Properties' and difficulty='700+' in the Drill Engine.
Good luck,
-Patrick
- Check out my site: GMATFix.com
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Patrick_GMATFix wrote:Hi Skalevar,skalevar wrote:If n and m are positive integers what is the remainder when 3^(4n +2) + m is divided by 10?
1) n = 2
2) m =1
The key is to solving this Q quickly is to come up with a strong rephrase. Recognize 2 general rules:
1) remainder when an integer is divided by 10 is the units digit of that integer. So the question can be rephrased as: What is the units digit of 3^(4n+2) + m?
2) units digit of an exponential expression follows a cyclical pattern as the exponent is incremented
In this case rephrasing is key. Since 3^(4n+2) is 9^(2n+1). In other words, it is 9 raised to an odd exponent. Regardless of the exponent, the units digit of 9^(odd) is 1. So to determine the units digit of 3^(4n+2) + m, we only need the units digit of m.
Rephrase: What is the units digit of m?
Statement 2 gives us this info so it's sufficient. The answer is B.
A more detailed explanation and a step-by-step video solution is available at GMATPrep Question 1395. If you struggle with advanced remainder problems, set topic='Number Properties' and difficulty='700+' in the Drill Engine.
Good luck,
Patrick
PLease explain. Regardless of the exponent, the units digit of 9^(odd) is 1. So to determine the units digit of 3^(4n+2) + m, we only need the units digit of m..
Regards
Deeyah
