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

Question

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.
(2) The remainder when n is divided by 10 is 3.

niks18 · Answer

\frac{m^2-n^2}{10}=\frac{(m+n)(m-n)}{10} . we need to know the value of  m  &  n  to get the remainder

Statement 1:  implies  m=10k+3 . but nothing mentioned about  n .  Insufficient

Statement 2:  implies  n=10q+3 . but nothing mentioned about  m .  Insufficient

Combining 1 & 2:  we have  m-n=10k+3-10q-3=10(k-q)  which is divisible by  10 . Hence remainder will be  0 .  Sufficient

Option  C

chetan2u · Answer

Note :- Remainder when div by 10 means the units digit of the number.

m^2-n^2...(m-n)(m+n) ..
Let's see the statements..

1) The remainder when m is divided by 10 is 3.
This means units digit of m is 3, but nothing about n.
Insufficient

(2) The remainder when n is divided by 10 is 3. 
This means units digit of n is 3, but nothing about m.
Insufficient

Combined..
Units digit of m and n both are 3, so m^2-n^2 will lead to units digit as 9-9=0
So it will be div by 10..
C

Kinshook · Answer

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

GMATWhizTeam · Answer

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

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

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