The remainder when m + n is divided by 12 is 8, and the remainder when

Question

The remainder when m + n is divided by 12 is 8, and the remainder when m - n is divided by 12 is 6. If m > n, then what is the remainder when mn divided by 6?

(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

aserghe1 · Accepted Answer

Since the remainder when m + n is divided by 12 is 8, the lowest value of m + n is 8. So we can write m+n=8. By the same logic, we can write m-n=6. Solving the 2 equations yields m=7, n=1. The remainder of 7*1/6 is 1.

Answer: A

Answer: A

chetan2u · Answer

Easiest and best way would be as also shown above:-
take the least possible values of m and n 
so  m+n=8  And  m-n=2 ..
Add two equations..  m+n+m-n=8+6....2m=14...m=7 
So n = 1..
 mn=7*1=7 ...
Thus remainder will be 1 when 7 is divided by 6..

Now a proper understanding.. 
 If m+n leaves a remainder of 8 when divided by 12, it will leave a remainder of 8-6=2 when divided by 6..
Similarly m-n leaves a remainder of 6 when divided by 12, it will leave a remainder of 6-6=0 when divided by 6.. 
So m and n leave the same remainder when divided by 6..
But m+n leaves a remainder of 2, so only possiblity is when both m and n leave a remainder of 1 each..

So mn will leave a remainder of 1*1=1
A

kunalcvrce · Answer

m+n=12k+8
m-n=12l+6

using these eq 
 m=6(some contant)+7
n=6(some contant)+1
m*n=(6(a)+1)*(6(b)+7)

by remainder theorem remainder is 1

option A

Funsho84 · Answer

Provided by the question:

1) (m+1)/ 12 = q+8
2) (m-n)/ 12 = q+6
3) m>n

rearranging the equations:
1) m+n = 12q + 8 
2) m-n = 12q + 6

Subtracting equation 1 from equation 2, you get 2n = 2 or n=1.

Plugging n=1 back in both equations, you get

m = 12q+ 7.

Now, if q=0, then m=7.

Now you have both values for n and m. 
n*m = 1*7 = 7.

7/6 will have a remainder of 1.

Answer A

gracie · Answer

another approach:
let difference between quotients=1
equation 1: m+n-8=12q
equation 2: m-n-6=12(q-1)=12q-12
subtracting equation 2 from 1,
2n-2=12
n=7
if q-1=0, then q=1
m+7-8=12*1
m=13
13*7=91
91/6 gives a remainder of 1
A

saraheja · Answer

(m+1)/ 12 = q+8
(m-n)/ 12 = q+6
m>n

m+n = 12q + 8 
m-n = 12q + 6

2n = 2 or n=1.

m = 12q+ 7.

n*m = 1*7 = 7.

7/6 will have a remainder of 1.

SaGa · Answer

(m+1)/ 12 = q+8
(m-n)/ 12 = q+6
m>n

m+n = 12q + 8 
m-n = 12q + 6

2n = 2 or n=1.

m = 12q+ 7.

n*m = 1*7 = 7.

7/6 will have a remainder of 1.

SidKan · Answer

Since the remainder when m + n is divided by 12 is 8, the lowest value of m + n is 8. So we can write m+n=8. By the same logic, we can write m-n=6. Solving the 2 equations yields m=7, n=1. The remainder of 7*1/6 is 1.

Answer: A

eswarchethu135 · Answer

m + n = 12a + 8
m - n = 12b + 6

Adding both,

2m = 12 (a + b) + 14
m = 6 (a + b + 1) + 1 (From this we can say m is a number, which when divided by 6 leaves a remainder of 1)

Subtracting both,

2n = 12 (a - b) + 2
n = 6 (a - b) + 1 (From this we can say n is a number, which when divided by 6 leaves a remainder of 1)

When m and n are multiplied and divided by 6, we get a remainder of 1 for each number. Hence the remainder when mn is divided by 6 is 1.

OPTION:  A

BrushMyQuant · Answer

The remainder when m + n is divided by 12 is 8

Theory: Dividend = Divisor*Quotient + Remainder

m + n -> Dividend
12 -> Divisor
a -> Quotient (Assume)
8 -> Remainder
=> m + n = 12*a + 8 = 12a + 8 ...(1)

The remainder when m - n is divided by 12 is 6

Let Quotient be b

=> m - n = 12b + 6 ...(2)

Adding (1) and (2) we get

m + n + m - n = 12a + 8 + 12b + 6 = 12*(a+b) + 14
=> 2m = 12*(a+b) + 14
=> m = 6*(a+b) + 7 = 6*(a+b+1) + 1

(1) - (2) we get

m + n -( m - n) = 12a + 8 -( 12b + 6) = 12*(a-b) + 2
=> 2n = 12*(a-b) + 2
=> n = 6*(a-b) + 1

=> m*n = (6*(a+b+1) + 1) * (6*(a-b) + 1)
=> on right hand side all terms will be a multiple of 6 except 1*1

=> m*n = multiple of 6 + 1
=> m*n will give 1 remainder when divided by 6

So,  Answer will be A 
Hope it helps!

Watch the following video to learn the Basics of Remainders

https://www.youtube.com/watch?v=P0S6j2uTaj4

plpurva · Answer

But the question did not state that the quotient for both equations is equal, why did we assume that?

chattyyee · Answer

Hello plpurva, Please Find Attached the solution without both being the same. https://gmatclub.com/forum/download/file.php?id=85979 GMAT-Club-Forum-8ocbtfl4.jpeg

aliasipsa · Answer

if I choose m+n=56 it is divisible by 12 (12*4=48) 56-48=8
56 can be 49+7, (49-7)/12=3 with 6 as a remainder 
m*n=49*7=329 329/6=54 and 5 as a remainder. How is it possible?

Bunuel · Answer

49 * 7 = 343, not 329.

aliasipsa · Answer

thank you, I am sorry

The remainder when m + n is divided by 12 is 8, and the remainder when

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The remainder when m + n is divided by 12 is 8, and the remainder when

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