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Thread: GCDs and LCMs

  1. #1
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    GCDs and LCMs

    Natural numbers $\displaystyle a,b,c$ are such that :

    i) GCD of $\displaystyle a,b$ is $\displaystyle 2^33^25^47^1$
    ii) GCD of $\displaystyle b,c$ is $\displaystyle 2^43^25^5$
    iii) GCD of $\displaystyle a,c$ is $\displaystyle 2^33^25^4$
    iv) Product of abc is $\displaystyle 2^1^13^1^05^1^77^2$

    Then find the least common multiple (LCM) of $\displaystyle a,b,c$.

    please help me. thank you.
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  2. #2
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    Quote Originally Posted by amey View Post
    Natural numbers $\displaystyle a,b,c$ are such that :

    i) GCD of $\displaystyle a,b$ is $\displaystyle 2^33^25^47^1$
    ii) GCD of $\displaystyle b,c$ is $\displaystyle 2^43^25^5$
    iii) GCD of $\displaystyle a,c$ is $\displaystyle 2^33^25^4$
    iv) Product of abc is $\displaystyle 2^1^13^1^05^1^77^2$

    Then find the least common multiple (LCM) of $\displaystyle a,b,c$.

    please help me. thank you.
    It's obvious (I hope) that each of the numbers a, b, c must be of the form $\displaystyle 2^?3^?5^?7^?$, and that the lcm will be the product of those four primes each raised to the highest of the powers that occur in a, b or c.

    You can deal with each of those four primes separately. Take 7 first, because that's the easiest. Each of a, b contains a 7 (to some nonzero power), but the product abc only contains 7 to the power 2. Therefore a, b each contain 7 to the power 1, and c does not contain 7 at all. So the highest power of 7 in the three numbers is $\displaystyle 7^1$, and hence the lcm will have a factor of 7.

    Now think about the powers of 3. Each pair of numbers has $\displaystyle 3^2$ in its gcd. So each of the three numbers contains 3 to the power at least 2. But if any pair of numbers contained 3 to a power greater than 2 then so would the gcd of those two numbers. Hence two of the numbers a, b, c must contain 3 to the power exactly 2. But the product abc contains $\displaystyle 3^{10}$. So if two of the numbers have $\displaystyle 3^2$ then the third one must have $\displaystyle 3^6$ (because 2+2+6=10). Thus the lcm must have 3 to the power 6.

    Now do a similar analysis of the powers of 2 and 5, and you will find that the lcm is $\displaystyle 2^?3^65^?7^1$ (where I have left the two ?s for you to find).
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  3. #3
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    Thanks. I got the remaining ones as $\displaystyle 2^4$ and $\displaystyle 5^8$.
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