Question: If m and n are positive integers, is the remainder of \(\frac{10^m+n}{3}\) greater than the remainder of\(\frac{10^n+m}{3}\) ?

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

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are NOT sufficient.

“If m and n are positive integers, is the remainder ofgreater than the remainder of?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

Answer:

Approach Solution (1):

First of all any positive integer can yield only three remainders upon division by 3: 0, 1, and 2

Since, the sum of the digits \(10^m and 10^n\) of is always 1 then the remainders of \(\frac{10^m+n}{3}\) and are only dependent on the value of the number added to \(10^m and 10^n\). There are three cases:
If the number added to them is: 0, 3, 6, 9, … then the remainder will be 1 (as the sum of the digits of \(10^m and 10^n\) will be 1 more than a multiple of 3);
If the number added tot hem is: 1, 4, 7, 10, …. Then the remainder will be 2 (as the sum of the digits of \(10^m and 10^n\) will be 2 more than a multiple of 3);
If the number added to them is: 2, 5, 8, 11, … then the remainder will be 0 (as the sum of the digits of \(10^m and 10^n\) will be a multiple of 3).

(1) m > n. Not sufficient
(2) The remainder of \(\frac{n}{2}\) is 2. So n could be: 2, 5, 8, 11, ... so we have the third case which means that the remainder of \(\frac{10^m+n}{3}\) is 0. Now, the question asks whether the remainder of \(\frac{10^m+n}{3}\) , which is 0, is greater than the remainder of \(\frac{10^m+n}{3}\) , which is 0, 1, or 2. Obviously it cannot be greater, it can be less than or equal to. So, the answer to the question is NO. Sufficient

Correct option: B

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