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Math Help - LCM (Least Common Multiple)

  1. #1
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    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.
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  2. #2
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    Quote Originally Posted by Frank Chan View Post
    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.
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  3. #3
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    LCM

    Hi Isomorphism,

    Thank you for your reply, can you tell me what is gcd stands for by return? Thanks again.
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  4. #4
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    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.
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  5. #5
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    dont understand last partwh

    what does d/y mean??
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  6. #6
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    Quote Originally Posted by blakelively07 View Post
    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.


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  7. #7
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    thanx

    thanx
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