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

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

M17-08

jpr200012 · Answer

To determine if a number is divisible by 3, add the digits and if they are divisible by 3, then the entire number is also.

We are given m and n are positive integers. The sum of the digits of 10 to any positive power will always be 1.

From statement 2, we are told the remainder of  n/3  will be 2.

For  (10^m + n)/3 , the  10^m  part will always give you 1 in the sum of the digits, and the n part will always give you 2 in the digits of the numerator. Therefore, this will always be divisible by 3 (have a remainder of 0).

The question asked if (10^m + n)/3 will ever have a larger remainder than  (10^n + m)/3 . Remainders can't be negative, therefore we know it will never be larger. There is no need to evaluate possible remainders of  (10^n + m)/3 .

Bunuel · Answer

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.

catty2004 · Answer

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

KarishmaB · Answer

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

raingary · Answer

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.

KarishmaB · Answer

The question is: 
is the remainder of 'a' larger than the remainder of 'b'?

We found that the remainder of 'a' is 0 which is the smallest possible remainder. No matter what the remainder of 'b' is, it will never be less than 0. So remainder of 'a' can never be larger than the remainder of 'b'. We can answer the question with 'No'. Therefore, statement 2 is sufficient.

nomis · Answer

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.

KarishmaB · Answer

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'.

newyork2012 · Answer

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

Bunuel · Answer

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.

AlwaysImproving · Answer

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.

KarishmaB · Answer

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.

road2gmat · Answer

I just came across this question, but I am having a hard time understanding the explanation. Why do we assume that "the sum of the digits of 10^m and 10^n is always equal to 1"? Is this a general rule or is it implied from the problem in any way?

Thank you in advance!

Thank you in advance!

Bunuel · Answer

10^m, where m is a positive integer, is a 1 followed by m zeros. Therefore, the sum of the digits of 10^m is always 1. For example, 10^1 = 10, 10^2 = 100, 10^3 = 1,000, and so on. The sum of the digits in each case is 1.

ManifestDreamMBA · Answer

Edit:  I have answer to the question I posted before, in case someone else also had the same question.

The trick to the question is to remember it's about the  remainder.  After we get this: is (1+n)/3 > (1+m)/3, we need to consider n/3 and m/3 values. here's why.

Say n = 2 and m = 1, the respective remainders become 0 and 1, which makes R(1+n)/3 < R(1+m)/3. 
 However if n = 1 and m = 0, then the remainders become 2 and 1, which makes R(1+n)/3 > R(1+m)/3

Hope this helps!

Initial Q 
I have a doubt. Wouldn't first express be a remainder of n+1 using binomial? Say (9+1)^m gives a remainder 1 and n is getting added, so the remainder is 1+n. Similarly the next expression gives a remainder of 1+m. Therefore if I know m>n, I know the remainder of second expression is greater than the first? I am trying to understand the gap in this approach.

ArpitaG · Answer

On further simplifying the question, we get is (1+n)/3 > (1+m)/3. Cant we further simplify the question stem to Is n > m using following 2 steps - (a) multiply both sides by 3, and (b) subtracting 1 from both sides? Accordingly, statement 1 also becomes sufficient since it answers in sufficiently no.

ManifestDreamMBA · Answer

The trick to the question is to remember it's about the  remainder.  After we get this: is (1+n)/3 > (1+m)/3, we need to consider n/3 and m/3 values. here's why.

Say n = 2 and m = 1, the respective remainders become 0 and 1, which makes R(1+n)/3 < R(1+m)/3. 
However if n = 1 and m = 0, then the remainders become 2 and 1, which makes R(1+n)/3 > R(1+m)/3

Hope this helps!

MagicMeat · Answer

rem (10^m)+n/3 will be (3^2+1)^m + n /3;; ie rem will be 1+n/3: n can be 0, 1;; similarly rem of (10^n)+m/3 will be 1+m/3: m can be 0, 1. the question is asking if 1+m/3 is greater than 1+n/3;; ie: if m>n ... this is my understanding, could someone please debunk why this wont be correct?

Bunuel · Answer

The issue is that remainders do not preserve the size relationship between two numbers. Even if m > n, the remainder when m is divided by 3 does not have to be greater than the remainder when n is divided by 3.

For example, if m = 4 and n = 2, then m > n, but the remainders are 1 and 2, respectively.

Also, the remainder when either m or n is divided by 3 can be  0, 1, or 2 , not just  0 or 1 .

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

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards