# Thread: LCM (Least Common Multiple)

1. ## LCM (Least Common Multiple)

Can someone help me in resolving the following problem?

Find all the solutions in positive integers for the equation x-y^4 =LCM(x,y).

Thank you.

2. Originally Posted by Frank Chan
Can someone help me in resolving the following problem?

Find all the solutions in positive integers for the equation x-y^4 =LCM(x,y).

Thank you.
Let $d = \text{gcd}(x,y)$ , then $\text{lcm}(x,y) = \frac{xy}{d}$

Suppose there exists a positive pair of solutions x and y, note that the following must hold:

$x - y^4 = \frac{xy}{d} \implies dx - dy^4 = xy \implies x(d - y) = dy^4$

By assumption on positivity of x and y, $d - y > 0 \implies y < d$, but since $d|y$ (by definition), $d \leq y$...Contradiction!

Thus there are no positive solutions x and y to the given equation.

3. ## LCM

Hi Isomorphism,

Thank you for your reply, can you tell me what is gcd stands for by return? Thanks again.

4. ## LCM

Hi Isomorphism,

I just find out that gcd is another term of hcf. Hence, I understand the whole solution now. Thank you for your great work, cheers.

5. ## dont understand last partwh

what does d/y mean??

6. Originally Posted by blakelively07
what does d/y mean??
It's not d/y. It's d|y (with a straight line). It means that d divides into y evenly, or that y divided by d gives you no remainder.

01

thanx