# Difficult Number System Solved QuestionAptitude Discussion

 Q. 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

Solution:
Option(A) is correct

Since the remainder when $m + n$ is divided by 12 is 8, $m + n = 12p + 8$;
and since the remainder when $m – n$ is divided by 12 is 6, $m – n = 12q + 6$.

Here, $p$ and $q$ are integers. Adding the two equations yields $2m = 12p + 12q + 14$.
Solving for $m$ yields $m = 6p + 6q + 7 = 6(p + q + 1) + 1 = 6r + 1$,
where $r$ is a positive integer equalling $p + q + 1$.

Now, let’s subtract the equations $m + n = 12p + 8$ and $m – n = 12q + 6$.
This yields $2n = (12p + 8) – (12q + 6) = 12(p – q) + 2$.

Solving for $n$ yields $n = 6(p – q) + 1 = 6t + 1$,
where $t$ is an integer equalling $p – q$

Hence, we have $mn = (6r + 1)(6t + 1)= 36rt + 6r + 6t + 1 = 6(6rt + r + t) + 1$ by factoring out 6

Hence, the remainder is 1.

## (2) Comment(s)

Shubham Garg
()

$m+n=12p+8$ and $m-n=12q+6.....$

Square both and subtract 2nd from 1st...You directly get the value of mn.....Then check the remainder
.

Shubham Garg
()

$m+n=12p+8$ and $m-n=12q+6.....$

Square both and subtract 2nd from 1st...You directly get the value of mn.....

Then check the remainder.