m and n are positive integer. Is the remainder of [(10^m)+n]/3 greater than the remainder of [(10^n)+m]/3?
(1) m>n
(2) The remainder of n/3 is 2
OA=B
If m m and n n are positive integers, is the remainder of
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The remainder of (10^p)/3 is always 1, irrespective of the values of p; where p is a positive integer.ziyuenlau wrote:m and n are positive integer. Is the remainder of [(10^m)+n]/3 greater than the remainder of [(10^n)+m]/3?
(1) m>n
(2) The remainder of n/3 is 2
OA=B
Thus the rephrased question is: Is the remainder of [1 + n]/3 greater than the remainder of [1 + m]/3?
Statement 1: m > n
Case 1: m = 3 and n = 2
The remainder of [1 + n]/3 = [1 + 2]/3 = 0
The remainder of [1 + m]/3 = [1 + 3]/3 = 1.
0 < 1. The answer is No.
Case 2: m = 5 and n = 3
The remainder of [1 + n]/3 = [1 + 3]/3 = 1
The remainder of [1 + m]/3 = [1 + 5]/3 = 0.
1 > 0. The answer is Yes. No unique answer. Insufficient.
Statement 2: The remainder of n/3 is 2
Say n = 3x + 2
Thus, the remainder of [1 + m]/3 = [1 + 3x + 2]/3 = [3x + 3] / 3 = 3(x+1)/3 = 0; since the numerator 3(x+1) is a factor of 3.
Since any number divisible by 3 can have 0, 1, or 2 as its remainder, and the remainder of [1 + n]/3 = 0, the remainder of [1 + m]/3 CANNOT be less than 0.
Or, in other words, the remainder of [(10^m)+n]/3 is NOT greater than the remainder of [(10^n)+m]/3. Sufficient.
The correct answer: B
Hope this helps!
Relevant book: Manhattan Review GMAT Data Sufficiency Guide
-Jay
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When a POWER OF 10 is divided by 3, the remainder will always be 1.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
10/3 = 3 R1.
100/3 = 33 R1.
1000/3 = 333 R1.
From there, the remainders will proceed in the following CYCLE: 2, 0, 1...2, 0, 1...2, 0, 1...
To illustrate:
(10 + 1)/3 = 11/3 = 3 R2.
(10 + 2)/3 = 12/3 = 4 R0
(10 + 3)/3 = 13/3 = 4 R1.
(10 + 4)/3 = 14/3 = 4 R2.
(10 + 5)/3 = 15/3 = 5 R0.
(10 + 6)/3 = 16/3 = 5 R1.
(100 + 1)/3 = 101/3 = 33 R2.
(100 + 2)/3 = 102/3 = 34 R0
(100 + 3)/3 = 103/3 = 34 R1.
(100 + 4)/3 = 104/3 = 34 R2.
(100 + 5)/3 = 105/3 = 35 R0.
(100 + 6)/3 = 106/3 = 35 R1.
Notice that the cycle of remainders is the SAME whether integer values are added to 10 or to 100.
Thus, the question stem can be rephrased in terms of 10:
Is the remainder of (10 + n)/3 larger than the remainder of (10 + m)/3?
Statement 1: m>n
Case 1: m=5 and n=2
(10 + n)/3 = 12/3 = 4 R0.
(10 + m)/3 = 15/3 = 5 R0.
Here, the remainders are EQUAL, so the answer to the question stem is NO,
Case 2: m=5 and n=1
(10 + n)/3 = 11/3 = 3 R2.
(10 + m)/3 = 15/3 = 5 R0.
Here, the first remainder of (10 + n)/3 is GREATER, so the answer to the question stem is YES.
INSUFFICIENT.
Statement 2: The remainder of n/3 is 2
Options for n are 5, 8, 11, 14...
If n=5, then (10 + n)/3 = 15/3 = 5 R0.
If n=8, then (10 + n)/3 = 18/3 = 6 R0.
If n=11, then (10 + 11)/3 = 21/3 = 7 R0.
The examples above illustrate that -- in every case -- the remainder of (10 + n)/3 will be 0.
Thus, the answer to the question stem is NO:
Since the remainder of (10 + n)/3 is 0, it is not possible that the remainder of (10 + n)/3 is larger than the remainder of (10 + m)/3.
SUFFICIENT.
The correct answer is B.
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Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.
As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.
For more information, please email me (Mitch Hunt) at [email protected].
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