If m and n are positive integers, and the remainder when m is divided

Question

If m and n are positive integers, and the remainder when m is divided by n is equal to the remainder when n divided by m, then which of the following could be the value of m*n?

I. 12
II. 24
III. 36

A. I only
B. II only
C. III only
D. I and III only 
E. II and II only

M40-03

Bunuel · Accepted Answer

If  m  and  n  are positive integers, and the remainder when  m  is divided by  n  is equal to the remainder when  n  is divided by  m , then which of the following could be the value of  m*n ?
 
I. 12
 
II. 24
 
III. 36

A. I only
B. II only
C. III only
D. I and III only
E. II and II only

We are given that the remainder when  m  is divided by  n  is equal to the remainder when  n  is divided by  m . This implies that  m  must be equal to  n , and consequently,  mn  must be the square of an integer. Therefore, the answer is C, III only.
 
To further elaborate on why  m  must be equal to  n , assume  m < n . In this case, the remainder when  m  is divided by  n  would be  m  itself. However, since we are given that the remainder when  m  is divided by  n  is equal to the remainder when  n  is divided by  m , it would mean the remainder when  n  is divided by  m  should also be  m . This scenario is impossible because the divisor must always be greater than the remainder. A similar contradiction arises if we assume  n < m . Therefore, the only consistent possibility is that  m  equals  n .

Answer: C

road740 · Answer

I´d say C, because the factorization of 36 can be 6*6, and so when m=n, then m/n and n/m will have the same remainder.

I tried out the factorizations of the other numbers as well, but it didn´t work.

JerryAtDreamScore · Answer

Breaking Down the Info:

the remainder when m is divided by n is equal to the remainder when n divided by m :

This only happens when  m = n . Then  m*n = m^2  must be a square. Hence the only viable option is III.

Answer: C

A small proof for the conclusion of  m = n :

Assume m > n on top of the given condition. Then  \frac{n}{m}  has a remainder of n. By the given condition,  \frac{m}{n}  has a remainder of n, which is as good as saying  \frac{m}{n}  has a remainder of 0. Then m > n cannot be true in this case. Similarly, m < n is also not true. Then we must have m = n.

unraveled · Answer

If m and n are positive integers, and the remainder when m is divided by n is equal to the remainder when n divided by m, then which of the following could be the value of m*n?

I. 12
II. 24
III. 36

A. I only
B. II only
C. III only
D. I and III only 
E. II and II only

Let's try each
m*n = 12
As each of m and n are positive integers, both can hold values that are factors of 12 i.e. 1,2, 3,4 ,6,12.
Since remainder of extreme values when divided by each other would not be equal, the middle value that re closest to each other would most satisfy the condition.
In this case 3 and 4 are possibilities but remainder is not equal. Hence this is not the option.

A and D are out.

m*n = 24
1,2,3, 4,6 ,8,12,24
Here 4 and 6 are possibilities but just like above example this is not satisfying the condition.

POE helps us hence answer is C.

Let's check anyway
m*n = 36
1,2,3,4, 6 ,9,12,18,36
Hence m and n are equal i.e. 6

Answer C.

Pif96 · Answer

I can't understand how it is a low difficult question

If m and n are positive integers, and the remainder when m is divided

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GMAT Problem Solving (PS) Questions

If m and n are positive integers, and the remainder when m is divided

Prep Toolkit

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