Answer to Question #165643 in Math for Hiroshima

Question

Determine if the statement is TRUE or FALSE. Justify your answer. All numbers under discussion are

integers.

5. For each m ≥ 1 and n ≥ 1, if mn is a multiple of 4, then m or n is a multiple of 4.

6. For each m ≥ 1 and n ≥ 1, if mn is a multiple of 3, then m or n is a multiple of 3

Answer 1

5) FALSE

Let $m=2$ and $n=2$ .

Number $mn=2\cdot 2=4$ is a multiple of $4$ , but $m$ and $n$ aren’t multiples of $4$ .

6) TRUE

Suppose to the contrary that $m$ and $n$ are not multiples of 3. 

This leads to three cases:

1.$\ m=3k+1,\ n=3l+1$

$mn=(3k+1)(3l+1)=9kl+3k+3l+1=3(3kl+k+l)+1$

The remainder on dividing $mn$ by $3$ equals $1$ .

2.$\ m=3k+1,\ n=3l+2$ $mn=(3k+1)(3l+2)=9kl+6k+3l+2=3(3kl+2k+l)+2$

The remainder on dividing $mn$ by $3$ equals $2$ .

3.$\ m=3k+2,\ n=3l+2$ $mn=(3k+2)(3l+2)=9kl+6k+6l+1=3(3kl+2k+2l+1)+1$

The remainder on dividing $mn$ by $3$ equals $1$ .

In each of cases, we have that $mn$ is not a multiple of 3. It follows that if $m$ and $n$ are not multiples of $3$, then $mn$ is not a multiple of $3$.

Therefore,  if $mn$ is a multiple of $3$ , then $m$ or $n$  is a multiple of $3$ .