The positive integers m and n leave remainders of 2 and 3, respectively, when divided by 6. m > n.
What is the remainder when m – n is divided by 6?
a)
1
b)
2
c)
3
d)
4
e)
5
Correct answer is option 'E'. Can you explain this answer?

Question

The positive integers m and n leave remainders of 2 and 3, respectively, when divided by 6. m > n.
What is the remainder when m – n is divided by 6?

a)
1
b)
2
c)
3
d)
4
e)
5

Correct answer is option 'E'. Can you explain this answer?

Answer

The minimum values for m and n are 14 and 9 respectively, with m being greater than n. Therefore, the difference between m and n is 5. In other words, when we divide the difference between m and n by 6, the result is 5. Hence, the correct answer is (E) 5.

Answer

If two numbers leave remainders of 2 and 3 when divided by 6, then we can write them as $6a+2$ and $6b+3$, where $a$ and $b$ are integers. We are given that $m>n$, so we can write $m=6a+2$ and $n=6b+3$.

The problem asks us to find the remainder when $m-n$ is divided by 6. We can substitute the expressions for $m$ and $n$ to find:
\begin{align*}
m-n &= (6a+2)-(6b+3)\
&= 6a-6b+2-3\
&= 6(a-b)-1.
\end{align*}Since $a$ and $b$ are integers, $a-b$ is also an integer. Therefore, $6(a-b)$ is a multiple of 6. The remainder when $6(a-b)$ is divided by 6 is $\boxed{0}$.

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