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Is statement

$$ lcm(ax, ay) = a *lcm(x,y) $$

true? (assuming all variables are positive integers)

I would say yes, but I can't find any confirmation or counter-example.

Ecto
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1 Answers1

1

It is true.

$$\text{lcm}(ax,ay)=\frac{|a^2xy|}{\text{gcd}(ax,ay)}.$$

Since $a,x,y \in \mathbb{Z}^{+}$ and $\text{gcd}(ax,ay)=a\cdot\text{gcd}(x,y)$,

$$=\frac{a^{2}xy}{a\cdot\text{gcd}(x,y)}=a\cdot\frac{xy}{\text{gcd}(x,y)}=a\cdot\text{lcm}(x,y).$$

Therefore, for $a,x,y \in \mathbb{Z}^{+}$, $\text{lcm}(ax,ay)=a\cdot\text{lcm}(x,y)$.

Shambhala
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