If m and n are positive integers, is the remainder of (10^m + n)/3

Official Solution:


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

Note: This is a difficult question that requires careful reading and understanding of the solution. It's important to take your time and fully comprehend the reasoning behind each step in the solution. Don't hesitate to ask for clarification if needed.

Firstly, it is important to note that any positive integer can only have three possible remainders upon division by 3: 0, 1, or 2.

Given that the sum of the digits of \(10^m\) and \(10^n\) is always equal to 1, the remainders of \(\frac{10^m + n}{3}\) and \(\frac{10^n + m}{3}\) depend only on the value of the number added to \(10^m\) and \(10^n\). There are three possible cases:

• If the number added to \(10^m\) and \(10^n\) is a multiple of 3 (i.e., 0, 3, 6, 9, ...), then the remainder when \(10^m+n\) or \(10^n+m\) is divided by 3 will be 1. This is because the sum of the digits of \(10^m+n\) and \(10^n+m\) will be 1 more than a multiple of 3.

• If the number added to \(10^m\) and \(10^n\) is one more than a multiple of 3 (i.e., 1, 4, 7, 10, ...), then the remainder when \(10^m+n\) or \(10^n+m\) is divided by 3 will be 2. This is because the sum of the digits of \(10^m+n\) and \(10^n+m\) will be 2 more than a multiple of 3.

• If the number added to \(10^m\) and \(10^n\) is two more than a multiple of 3 (i.e., 2, 5, 8, 11, ...), then the remainder when \(10^m+n\) or \(10^n+m\) is divided by 3 will be 0. This is because the sum of the digits of \(10^m+n\) and \(10^n+m\) will be a multiple of 3.

(1) \(m \gt n\). Not sufficient.

(2) The remainder of \(\frac{n}{3}\) is \(2\).

The above implies that \(n\) can take any of the values 2, 5, 8, 11, and so on. When any such value of \(n\) is added to \(10^m\), the resulting number has digits that add up to a multiple of 3, which means that its remainder when divided by 3 is 0. Thus, 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 \(\frac{10^n + m}{3}\), which can only be 0, 1, or 2. It is clear that the remainder of \(\frac{10^m + n}{3}\) cannot be greater than the remainder of \(\frac{10^n + m}{3}\), it may only be less than or equal to it. Therefore, the answer to the question is NO. This statement alone is sufficient.

Answer: B

Hope it's clear.

crystal clear! Just hope that my brain will remember this on the test day!

You can also use binomial theorem here. Again, let me reiterate that there are many concepts which are not essential for GMAT but knowing them helps you get to the answer quickly.

The moment I see \(\frac{10^m + n}{3}\) here, my mind sees \(\frac{(9+1)^m + n}{3}\)
So I say that \(\frac{10^m}{3}\) and \(\frac{10^n}{3}\) give remainder 1 in any case (m and n are positive integers). I just need to worry about n/3 and m/3.

1. \(m \gt n\)
Doesn't tell me about the remainder when m and n are divided by 3.

2. The remainder of \(\frac{n}{3}\) is \(2\)
If n/2 gives a remainder of 2, total remainder of \(\frac{10^m + n}{3}\) is 1+2 = 3 which is equal to 0. So no matter what the remainder of \(\frac{m}{3}\), the remainder of \(\frac{10^n + m}{3}\) will never be less than 0. Hence sufficient.

Answer B

I have a doubt why B is sufficient to answer the question.
I understood the solution but if we take only the statement B then there is nothing to prove that m is not equal to n.
As in the given statement m and n are positive integers but what is the relation between them is not provided whether m >n , m<n or m=n .

Also ,by B we are only able to prove that remainder will not be lesser but what if equal or greater.

to rephrase the question is x greater than y ( here x and y are the remainders).
If x is greater than y, answer is yes but if x is not greater than y the answer is no.

If x is 0 and y is 0 ----No
If x is 1 or 2 and y is 0 , the Yes

Why is the answer not E?

If the question is reversed -- is the remainder of 10^n +m/3 larger than the remainder of 10^m+n/3 ? then the answer would be B as
0 can never be greater than 0, 1 or 2.

Because you find that a = 0 (the remainder of the first expression is 0). Hence under no condition can 'a' be greater than 'b'. Hence, it is sufficient to answer with an emphatic 'No'.

i don;t understand ....the answer is a NO ....why is the correct answer is B ? not E?

There are two types of data sufficiency questions:

1. YES/NO DS Questions:

In a Yes/No Data Sufficiency questions, statement(s) is sufficient if the answer is “always yes” or “always no” while a statement(s) is insufficient if the answer is "sometimes yes" and "sometimes no".


2. VALUE DS QUESTIONS:

When a DS question asks about the value of some variable, then the statement(s) is sufficient ONLY if you can get the single numerical value of this variable.

This question is a VALUE question, so a statement to be sufficient it should give the single numerical value of the units digit. As you said, from (1) it could be 1 or 6, so (1) is not sufficient.

Hope it's clear.

Hi VeritasKarishma,

The question didn't specify whether m=n. What if m=n? Doesn't the statement 1 hold relevance in such event?

Thanks.

Question:
Is remainder of x larger than the remainder of y?

We can answer this in two ways:
1. Yes. Remainder of x is larger than remainder of y.
2. No. Remainder of x is same as remainder of y. Or remainder of x is less than remainder of y.
If the remainders are equal or remainder of x is smaller, we will answer it with an emphatic "No" only. In both cases answer is "No". Hence it doesn't matter.
If remainder of x is 0, then it cannot be GREATER than remainder of y in any case. It may be equal or small. Hence, answer will remain "No".

So statement I is not needed.

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