If m and n are positive integers, what is the remainder when m^2 - n^2

Asked: If m and n are positive integers, what is the remainder when m^2 - n^2 is divided by 10?

(1) The remainder when m is divided by 10 is 3.
m = 10k + 3
Since n is unknown
NOT SUFFICIENT

(2) The remainder when n is divided by 10 is 3.
n = 10k + 3
Since m is unknown
NOT SUFFICIENT

(1) + (2)
(1) The remainder when m is divided by 10 is 3.
m = 10x + 3
(2) The remainder when n is divided by 10 is 3.
n = 10y + 3
m^2 - n^2 = {10(x+y) + 6}{10(x-y)} = 10(x-y)(10x + 10y + 6)
The remainder when m^2 - n^2 is divided by 10 = 0
SUFFICIENT

IMO C

Step 1: Analyse Question Stem

m and n are positive integers.
We have to find the remainder when \(m^2\) – \(n^2\) is divided by 10.

A number is divisible by 10 if the number’s unit digit is ZERO. If the unit digit is non-zero, the unit digit represents the remainder when the number is divided by 10.
Therefore, the question can be rephrased as “What is the units digit of the expression \(m^2\) – \(n^2\)?”

Step 2: Analyse Statements Independently (And eliminate options) – AD / BCE

Statement 1: The remainder when m is divided by 10 is 3

This means the unit digit of m is 3. Therefore, the possible values of m could be 3 or 13 or 23 and so on.
However, statement 1 does not provide any information about n. Therefore, the unit digit of \(m^2\) – \(n^2\) cannot be calculated.

The data in statement 1 is insufficient to find out the unit digit of \(m^2\) – \(n^2\).
Statement 1 alone is insufficient. Answer options A and D can be eliminated.

Statement 2: The remainder when n is divided by 10 is 3.

This means the unit digit of n is 3. Therefore, the possible values of n could be 3 or 13 or 23 and so on.
However, statement 2 does not provide any information about m. Therefore, the unit digit of \(m^2\) – \(n^2\) cannot be calculated.

The data in statement 2 is insufficient to find out the unit digit of \(m^2\) – \(n^2\).
Statement 2 alone is insufficient. Answer option B can be eliminated.

Step 3: Analyse Statements by combining

From statement 1: The unit digit of m is 3, so m = 3, 13, 23, …
From statement 2: The unit digit of n is 3, so n = 3, 13, 23,…..

Unit digit of \(m^2\) = 9
Unit digit of \(n^2\) = 9

Therefore, unit digit of \(m^2\) – \(n^2\) = 9 – 9 = 0, which means that the expression \(m^2\) – \(n^2\) is divisible by 10 and hence the remainder when it is divided by 10 is ZERO.

The combination of statements is sufficient to find a unique value for the remainder.
Statements 1 and 2 together are sufficient. Answer option E can be eliminated.

The correct answer option is C.

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