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remainder of (10^m + n)/3

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by arora007 » Tue Jul 06, 2010 3:43 am
If m and n are positive integers, is the remainder of (10^m + n)/3 larger than the remainder of (10^n + m)/3 ?

1. m > n
2. The remainder of n/3 is 2

OA is B, looking for an easy explanation.

the problem is from gmatclub tests, really good math out there...
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Source: — Data Sufficiency |
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by kvcpk » Tue Jul 06, 2010 4:01 am
m and n are positive integers
remainder of (10^m + n)/3 larger than the remainder of (10^n + m)/3 ?

m>n

put m=2,n=1
101/3 - remainder is 2
12/3 - reminder is 0
2>0

put m=4,n=1
10001/3 - reminder is 2
14/3 - reminder is 2
2=2
INSUFFICIENT

The remainder of n/3 is 2

reminder of [10^(some vale) + x] when divided by 3 is equal to 1 + (reminder of x/3)
why?
because reminder of 10^(somevalue)/3 is always 1

hence reminder of (10^m + n)/3 = 1+2=3 which means it is divisible by 3. Hence reminder is 0.

what is reminder of (10^n + m)/3 now?
Well, we need not calculate... because 0 cannot be larger than any other reminder..

pick B

Hope this helps!!
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by sanju09 » Tue Jul 06, 2010 4:29 am
arora007 wrote:If m and n are positive integers, is the remainder of (10^m + n)/3 larger than the remainder of (10^n + m)/3 ?

1. m > n
2. The remainder of n/3 is 2

OA is B, looking for an easy explanation.

the problem is from gmatclub tests, really good math out there...

If a is a positive integer, the remainder when 10^a is divided by 3, is always 1. Hence, the remainder of (10^m + n)/3 is same as the remainder of (1 + n)/3, and that the remainder of (10^n + m)/3 is same as the remainder of (1 + m)/3, for some positive integers m, n. Now, the question reduces to

Is the remainder of (1 + n)/3 greater than the remainder of (1 + m)/3?

(1) When m > n, take easy going numbers for m, n; like m = 2, n = 1, and the remainder of (1 + n)/3 comes out to be greater than the remainder of (1 + m)/3, but when m = 3, n = 2; the remainder of (1 + n)/3 comes out to be smaller than the remainder of (1 + m)/3. Insufficient

(2) If the remainder of n/3 is 2, 1 + n is a multiple of 3, and the remainder of (1 + n)/3 is 0. But we have no clue about how m/3 behaves. In other words, m could or could not behave like n, thus leaving more than one possibilities as an answer to the stem. Insufficient

When taken together, 1 + m is greater than a multiple of 3, and hence 1 + m would always leave more remainder than what a multiple of 3 leaves, when divided by 3, only if 1 + m is not a greater multiple of 3 than 1 + n.

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by kvcpk » Tue Jul 06, 2010 4:39 am
@Sanju -

You are right till the last step.. Let me explain what you missed.
You go the reminder of (1 + n)/3 as 0.

Now, What is the question asking us?

is the remainder of (10^m + n)/3 larger than the remainder of (10^n + m)/3 ?
-> is 0 larger than the remainder of (10^n + m)/3 ?

Can 0 be larger than some other reminder{0 or 1 or 2}? The answer is always NO.

hence pick B

hope this helps!!
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by sanju09 » Tue Jul 06, 2010 5:00 am
kvcpk wrote:@Sanju -

You are right till the last step.. Let me explain what you missed.
You go the reminder of (1 + n)/3 as 0.

Now, What is the question asking us?

is the remainder of (10^m + n)/3 larger than the remainder of (10^n + m)/3 ?
-> is 0 larger than the remainder of (10^n + m)/3 ?

Can 0 be larger than some other reminder{0 or 1 or 2}? The answer is always NO.

hence pick B

hope this helps!!
Oh yeah

Is the remainder of (1 + n)/3 greater than the remainder of (1 + m)/3?

can then be rephrased as

Is 0 greater than any of 0, 1, or 2?

And the answer is always NO!

Thanks for your reminder kvcpk :)
Last edited by sanju09 on Tue Jul 06, 2010 5:03 am, edited 1 time in total.
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by kvcpk » Tue Jul 06, 2010 5:02 am
sanju09 wrote:And the answer is always NO!

Thanks kvcpk
B-)
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