Guided Example #3 - Natural Numbers

One response: Let a and b represent the integers.

Let G represents the GCF of a and b.

Then ${\displaystyle a=Gx}$ and ${\displaystyle b=Gy}$ , where x and y are relatively prime.

The LCM of a and b is ${\displaystyle Gxy}$ .

Since ${\displaystyle x=\frac{a}{G}\text{and}y=\frac{b}{G}}$ , by substitution, ${\displaystyle Gxy=G\left(\frac{a}{G}\right)\left(\frac{b}{G}\right)}$ .

Hence, Gxy (or the LCM of a and b) = ${\displaystyle \frac{ab}{G}}$ .

One response: Let a and b represent the integers.

Let G represents the GCF of a and b.

Then ${\displaystyle a=Gx}$ and ${\displaystyle b=Gy}$ , where x and y are relatively prime.

The LCM of a and b is ${\displaystyle Gxy}$ .

Since ${\displaystyle x=\frac{a}{G}\text{and}y=\frac{b}{G}}$ , by substitution, ${\displaystyle Gxy=G\left(\frac{a}{G}\right)\left(\frac{b}{G}\right)}$ .

Hence, Gxy (or the LCM of a and b) = ${\displaystyle \frac{ab}{G}}$ .