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Qn: If the product of two integers is $2^7 \cdot 3^8 \cdot 5^2 \cdot 7^{11}$ and their greatest common divisor is $2^3 \cdot 3^4 \cdot 5$, what is their least common multiple?

I have issue with this question please help me solve it.

I tried assuming that lcm is $x$ =. Then, Gcd $\cdot x = 2^3 \cdot 3^4 \cdot 5x$. And, product factors /Gcd $x$

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marked as duplicate by Bill Dubuque prime-numbers May 11 at 15:02

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  • $\begingroup$ Hint: $lcm(a, b) = ab / gcd(a, b)$. $\endgroup$ – Borna Ghahnoosh May 11 at 14:54
  • $\begingroup$ How many 2's do you need in the prime factorization of lcm for it to be a multiple of both integers? How many 3's? How many 5's? How many 7's? $\endgroup$ – Dzoooks May 11 at 14:56
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The product of two integers is the gcd times the lcm. You can demonstrate that by considering the prime factorizations of each. The gcd gets the minimum exponent of the two numbers and the lcm gets the maximum.

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