If m and n are positive integers such that m n, what is the Re.

Question

If m and n are positive integers such that m > n, what is the remainder when m^2 – n^2 is divided by 21?

Statement 1: The remainder when (m + n) is divided by 21 is 1.

Statement 2: The remainder when (m – n) is divided by 21 is 1.

Abhishek009 · Answer

FROM STATEMENT - I     ( INSUFFICIENT)

We do not have any relationship between the value of m & n as such it won't be possible for us to find the remainder of  \frac{(m^2 – n^2)}{21} , since m & n can take any values...

FROM STATEMENT - I     ( INSUFFICIENT)

We do not have any relationship between the value of m & n as such it won't be possible for us to find the remainder of  \frac{(m^2 – n^2)}{21} , since m & n can take any values...

COMBINE STATEMENT I & II     (SUFFICIENT)

\frac{(m^2 – n^2)}{21} = \frac{( m+n )( m-n )}{21}  =  Remainder \ 1 
 
Hence BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked, answer will be (C)

Similar Question to practice  Here

AkshdeepS · Answer

Bunuel, pushpitkc : Please bring some light on this question using number plugging technique.

QZ

BrentGMATPrepNow · Answer

Given : m and n are positive integers such that m > n

Target question :    What is the remainder when m² – n² is divided by 21?

Statement 1 : The remainder when (m + n) is divided by 21 is 1 
There are several values of m and n that satisfy statement 1. Here are two: 
 Case a : m = 12 and n = 10. This means m + n = 12 + 10 = 22, and 22 divided by 21 leaves remainder 1. In this case, m² – n² = 12² – 10² = 144 - 100 = 44. 
When we divide 44 by 21, we get 2 with remainder 2. So, the answer to the target question is  when m² – n² is divided by 21, the remainder is 2 
 Case b : m = 13 and n = 9. This means m + n = 13 + 9 = 22, and 22 divided by 21 leaves remainder 1. In this case, m² – n² = 13² – 9² = 169 - 81 = 88. 
When we divide 88 by 21, we get 4 with remainder 4. So, the answer to the target question is  when m² – n² is divided by 21, the remainder is 4 
Since we cannot answer the  target question  with certainty, statement 1 is NOT SUFFICIENT

Statement 2 : The remainder when (m – n) is divided by 21 is 1 
There are several values of m and n that satisfy statement 2. Here are two: 
 Case a : m = 5 and n = 4. This means m - n = 5 - 4 = 1, and 1 divided by 21 leaves remainder 1. In this case, m² – n² = 5² – 4² = 25 - 16 = 9. 
When we divide 9 by 21, we get 0 with remainder 9. So, the answer to the target question is  when m² – n² is divided by 21, the remainder is 9 
 Case b : m = 4 and n = 3. This means m - n = 4 - 3 = 1, and 1 divided by 21 leaves remainder 1. In this case, m² – n² = 4² – 3² = 16 - 9 = 7. 
When we divide 7 by 21, we get 0 with remainder 7. So, the answer to the target question is  when m² – n² is divided by 21, the remainder is 7 
Since we cannot answer the  target question  with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined    
Statement 1 tells us that the remainder is 1 when (m + n) is divided by 21
In other words, m+n is  1 greater  than some multiple of 21. 
So, we can write:  m+n = 21k + 1  (for some integer k)

Statement 2 tells us that the remainder is 1 when (m - n) is divided by 21
In other words, m-n is  1 greater  than some multiple of 21. 
So, we can write:  m-n = 21j + 1  (for some integer j)

Now recognize that we can factor m² – n² 
We get: m² – n² = ( m + n )( m - n )
= ( 21k + 1 )( 21j + 1 )
= 21²mn + 21k + 21j + 1
= 21(21mn + k + j) + 1
Since 21(21mn + k + j) is definitely a multiple of 21, we can conclude that 21(21mn + k + j) + 1 is  1 greater  than some multiple of 21.
In other words, m² – n² is  1 greater  than some multiple of 21.
So,  when m² – n² is divided by 21, the remainder is 1 
Since we can answer the  target question  with certainty, the combined statements are SUFFICIENT

Answer: C

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fskilnik · Answer

Important: the solution presented below assumes a reasonable quantitative maturity. 
(If you do not have that, I recommend the excellent step-by-step presentation offered by Brent above.)

m > n \ge 1\,\,\,{\rm{ints}}\,\,\,\left( * \right)

{m^{\rm{2}}} - {n^2} = 21K + R

K,R\,\,{\rm{ints}}\,\,,\,\,\,0 \le R \le 20

? = R

\left( 1 \right)\,\,m + n = 21J + 1\,\,,\,\,\,J\mathop \ge \limits^{\left( * \right)} 1\,\,\,{\mathop{\rm int}}

\left\{ \matrix{
 \,{\rm{Take}}\,\,\left( {m,n} \right) = \left( {21,1} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{m^{\rm{2}}} - {n^2} = {21^2} - {1^2}\,\,\, \Rightarrow \,\,\,\,R = \,\,20 \hfill \cr 
 \,{\rm{Take}}\,\,\left( {m,n} \right) = \left( {20,2} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{m^{\rm{2}}} - {n^2} = {20^2} - {2^2} = \left( {20 - 2} \right)\left( {20 + 2} \right) = \left( {21 - 3} \right)\left( {21 + 1} \right) = 21\left( {21 + 1 - 3} \right) - 3\,\,\, \Rightarrow \,\,\,\,R = \,\,18\, \hfill \cr} \right.

\left( 2 \right)\,\,m - n = 21L + 1\,\,,\,\,\,L\mathop \ge \limits^{\left( * \right)} 0\,\,\,{\mathop{\rm int}} \,

\left\{ \matrix{
 \,{\rm{Take}}\,\,\left( {m,n} \right) = \left( {2,1} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{m^{\rm{2}}} - {n^2} = 3\,\,\, \Rightarrow \,\,\,\,R = \,\,3 \hfill \cr 
 \,{\rm{Take}}\,\,\left( {m,n} \right) = \left( {3,2} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{m^{\rm{2}}} - {n^2} = 5\,\,\, \Rightarrow \,\,\,\,R = \,5\, \hfill \cr} \right.

\left( {1 + 2} \right)\,\,\,\,\left( {21J + 1} \right)\left( {21L + 1} \right) = 21\left( {21JL + J + L} \right) + 1\,\,\, \Rightarrow \,\,\,\,R = \,\,1\,\,\,\,

This solution follows the notations and rationale taught in the GMATH method.

Regards, 
Fabio.

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If m and n are positive integers such that m > n, what is the Re.

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