LCM (Least Common Multiple)

**Frank Chan** · June 13th 2009, 08:52 PM

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.

**Isomorphism** · June 13th 2009, 11:18 PM

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.

**Frank Chan** · June 14th 2009, 01:21 AM

Hi Isomorphism,

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

**Frank Chan** · June 14th 2009, 01:27 AM

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.

**blakelively07** · June 25th 2009, 02:06 AM

what does d/y mean??

**yeongil** · June 25th 2009, 03:02 AM

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

**blakelively07** · June 27th 2009, 12:13 AM

thanx

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