Question

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

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

multiple of 4.

2. 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

1. To show that this statement is false, we look for a counterexample

$m=2\geq1, n=2\geq1$

$$mn=2(2)=4$$

Since 4 is a multiple of 4, then $mn$ is a multiple of 4. But $m=2$ is not a multiple of 4, $n=2$ is not a multiple of 4.

False.

2. $m$ is not a multiple of 3 and $n$ is not a multiple of 3.

Possible cases

$$m=3a+1, n=3b+1,$$ $$mn=(3a+1)(3b+1)=9ab+3a+3b+1$$

$mn$ is not a multiple of 3. Contradiction.

$$m=3a+1, n=3b+2,$$ $$mn=(3a+1)(3b+2)=9ab+6a+3b+2$$

$mn$ is not a multiple of 3. Contradiction.

$$m=3a+2, n=3b+1,$$ $$mn=(3a+2)(3b+1)=9ab+3a+6b+2$$

$mn$ is not a multiple of 3. Contradiction.

$$m=3a+2, n=3b+2,$$ $$mn=(3a+2)(3b+2)=9ab+6a+6b+4$$ $$=9ab+6a+6b+3+1$$

$mn$ is not a multiple of 3. Contradiction.

Therefore if mn is a multiple of 3, then $m\geq1$ or $n\geq1$ is a multiple of 3.

True.