One way I visualize GCD*LCM=a*b is by thinking about the two numbers' prime box (set of prime factors):
e.g.
70: 2 5 7
30: 2 3 5
GCD: pull out
mutual factors: 2 5; multiply = 10 = GCD
LCM: pull out
all factors and cross out duplicated ones: 2 5 7 X 3 X; multiply = 210 = LCM.
Notice the crossed-out factors in LCM are actually the ones in GCD. Thus, when you GCD * LCM, you reunite all factors, getting you back to a * b.
Hope it helps.
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3 Key Lessons Learned from My Journey from a 580 to a 740
Success springs not so much from talents as from consistency.
GMAT questions are to the mind what dumbbells are to the body.
Just as I like Quant, so (too) I enjoy Verbal, an interesting piece of the GMAT puzzle.
My GMAT skill in 2020 is higher than that in 2019. = My GMAT skill is higher in 2020 than (it was) in 2019. = My GMAT skill is higher in 2020 than was the case in 2019.