Divisibility (algebra, number theory).

Question

Suppose you have

$$a b c = m$$

$$a|y$$ $$b|y$$ $$c|y$$

Does that imply that

$$m|y$$

$$am_1 = y$$ $$bm_2 = y$$ $$cm_3 = y$$

$$mm_1 m_2 m_3 = y^3$$

??

Answer 1

No. Try $a=b=c=y=2$.

If $a,b,c$ are coprime, then yes.

Answer 2

$$a|y $$ $$b|y $$ $$c|y$$ imply $$\text {lcm}(a, b, c)|y$$ but not $$abc|y $$