I was going through the Hard Practice quiz and came across this question. I manipulated the choices and moved the GCF to quantity A. That gave us 2*LCM*GCF and noting the property that LCM*GCF= product of the two numbers, I thought this would just depend on the values for x and y. Since the question doesn’t explicitly say positive odd numbers, I considered the cases where they were both positive vs where they were negative. Turns out my answer was wrong. Can someone help me understand why?
Can you show this?
Quantity A Quantity B
2\*LCM(x, y) (x\*y)/GCF(x,y)
Step1 2*LCM(x, y)*GCF(x,y) x*y
Step2 2*x*y x*y
I thought x and y could be either both positive or both negative or one positive and one negative. In the third case the Quantity A would be less than B.
But I think LCM and GCF are always positive (just googled) and that’s why this won’t work. But this also means that the property LCM*GCF= a*b holds if both a and b are either positive or negative
well it’s technically \operatorname{LCM}(a,b) \cdot \operatorname{GCD} (a,b) = |a \cdot b|
Yes. As a side note, GCD can be 0 too
Not really, Quantity A would still be bigger. In essence, you’re really comparing 2 |xy| with xy.