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the least common multiple, pls help :(

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by [email protected] » Sun May 13, 2012 2:31 am
If a and b are positive even integers, and the least common multiple of a and b is expressed as (a*b)/n, which of the following statements could be false?

A n is a factor of both a and b
B (a*b)/n < ab
C ab is multiple of 2.
D (a*b)/n is a multiple of 2.
E n is a multiple of 4.

Can someone explain the answer? Even so I read the answer, I don't get it!














OA = E
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by mathbyvemuri » Sun May 13, 2012 4:04 am
As a and b are even integers, a and b are multiples of 2. and hence a*b is a multiple of 4 and option C is ruled out.
LCM of a and b must be multiple of 2 => option D is out
Take some examples:
2 and 6. LCM = 6, and this can be expressed as 2*6/2. here n=2
2 and 4. LCM = 4, and this can be expressed as 2*4/2. here n=2
6 and 36. LCM = 36, and this can be expressed as 6*36/6. here n=6
6 and 8. LCM = 24, and this can be expressed as 6*8/2. here n=2
12 and 16. LCM = 48, and this can be expressed as 12*16/4. here n=4
If one of a and b is a multiple of other, then n is the smallest of the two.
The minimum value of n is 2 as both the numbers are even=> a*b/n < ab => B is out
In any case n is a factor of a and b => a is out
But in all cases n is not a multiple of 4.
It is clear that, option E could be false
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