Confusing solution
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Module 5 Day 7 Your Turn Part 3 Mini-Question Solution
Let the GCD of a and b be g; then we can express a as g × A and b as g × B, where A and B don't have any factors in common. Then the LCM of a and b is g × A × B, and the LCM divided by the GCD would be A × B. Since this is equal to b, that means that g × B = A × B, which means that g = A. Then a = A^2, and b is a multiple of the square root of a.
Bold part is confusing -
@neatlobster (I'm gonna copy over the question first):
Following the explanation, we have the following:
Remember, the problem gave us that . Using our substitutions for the lcm & gcd, we have that
Now, here's the clever substitution: recall that . So,
Since , so is a perfect square.
And, since
, so is a multiple of .
Hence, the answer is the fifth one.I know this is a lot, let me know if you have any more questions. I'd be glad to help!
P.S. Here's a hint: try using . It's a very useful fact!
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Thanks so much!
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@neatlobster @quacker88 Thank you!!