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GMAT Focus Exponents/Remainder Question

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by bryan22583 » Sat Apr 13, 2013 12:42 pm
If K is a positive integer, what is the remainder when (13^4K+2) +8 is divided by 10?

A) 7
B) 4
C) 2
D) 1
E) 0

Answer:[spoiler] (A) 7 [/spoiler]




Official explanation requires one to calculate 13^4 = 28561, then multiply by 13^2 = 169, to conclude that the units digit must be 9, and thus when 8 is added the final units digit becomes 7, yielding a remainder of 7 when divided by 10. Is there a better way to do this problem?
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by Anju@Gurome » Sat Apr 13, 2013 12:50 pm
bryan22583 wrote:If K is a positive integer, what is the remainder when (13^4K+2) +8 is divided by 10?
We need to determine the units digit of 13^(4k + 2) + 8

Units digit of 13^1 = 3
Units digit of 13^2 = 9
Units digit of 13^3 = 7
Units digit of 13^4 = 1
Units digit of 13^5 = 3 and so on...

Note that we don't need to calculate the actual value of 13^4. We can just keep on multiplying the units digit with 3. Now you can follow the official explanation or you can notice that units digit of powers of 13 has a cycle of 3, 9, 7, 1, 3...
Hence, units digit of 13^(any multiple of 4 + 2) will be 9

Hence, units digit of 13^(4k + 2) + 8 = units digit of (9 + 8) = 7

The correct answer is A.
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by stpetrus2 » Sat Apr 13, 2013 1:57 pm
149 is a prime number -> 149 is the smallest number we can divide to and ends in 9 add the 8 and the number ends in 7 which is our remainder
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by stpetrus2 » Sat Apr 13, 2013 1:57 pm
149 is a prime number -> 149 is the smallest number we can divide to and ends in 9 add the 8 and the number ends in 7 which is our remainder
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by GMATGuruNY » Sat Apr 13, 2013 1:59 pm
bryan22583 wrote:If K is a positive integer, what is the remainder when (13^4K+2) +8 is divided by 10?

A) 7
B) 4
C) 2
D) 1
E) 0
Let k=1.
Then 13^(4K+2) + 8 = 13^(4*1 + 2) + 8 = 13� + 8.

When a positive integer is divided by 10, the remainder is the units digit of the integer:
12/10 = 1 R2.
587/10 = 58 R7.
And so on.

Thus, to determine the remainder when 13� + 8 is divided by 10, we need to know the units digit of 13� + 8.

Since 3� = 3² * 3� = 9*81 = 729, 13� has a units digit of 9.
Since 9+8 = 17, the units digit of 13� + 8 must be 7.
Thus, when 13� + 8 is divided by 10, the remainder will be 7.

The correct answer is A.
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