gettinit wrote:
VeritasPrepKarishma wrote:
rxs0005 wrote:
If the least common multiple of integers x and y is 840, what is the value of x?
(1) The greatest common factor of x and y is 56.
(2) y = 168
Remember a property of LCM and GCF:
If x and y are two positive integers, LCM * GCF = x* y
Stmnt 1: Just the GCF will give you the product of the two numbers. Not their individual values.
Stmnt 2: Knowing the value of y and LCM is not sufficient to get x.
x could be 840 or 105 or 5 etc.
Using both together, you get x = 840*56/168 = 280
Answer (C).
Karishma is this a known property? I have never ran across it before? I am not sure I understand why it works either. Can you please explain?
It is taught at school (though curriculums across the world vary)
Let us take an example to see why this works:
\(x = 60 = 2^2 * 3 * 5\)
\(y = 126 = 2*3^2*7\)
Now GCF here will be \(6 (= 2*3)\) (because that is all that is common to x and y)
LCM will be whatever is common taken once and the remaining i.e. \((2*3) * 2*5 * 3*7\)
When you multiply GCF with LCM, you get
\((2*3) * (2*3 *2*5 * 3*7)\) i.e. whatever is common comes twice and everything else that the two numbers had.
I can re-arrange this product to write it as \((2*3 * 2*5) * (2*3 * 3*7)\) i.e. 60*126
This is the product of the two numbers.
Since GCF has what is common to them and LCM has what is common to them written once and everything else, GCF and LCM will always multiply to give the two numbers.
Can you now think what will happen in case of 3 numbers?
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