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

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

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.