If a and b are two distinct positive integers such that b a, which

Question

If a and b are two distinct positive integers such that b > a, which of the following expressions is equivalent to the difference between the maximum and the minimum possible values of the ratio of the least common multiple of a and b to the highest common factor of a and b ?

A.  \frac{a}{b}*(a + 1)(a - 1)

B.  \frac{a}{b}*(b + 1)(b - 1)

C.  \frac{b}{a}*(a + 1)(a - 1)

D.  \frac{b}{a}*(b + 1)(b - 1)

E.  b*(a + 1)(a - 1)

RaghavKhanna · Accepted Answer

Max   = ab/1 , ie; a and b are co - primes.

Min   = b / a

Difference = ab - b/a = b(a^2-1)/a

Answer = C

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NitishJain · Answer

IMO  C

to find: Max ( \frac{LCM(a,b)}{HCF(a,b)} ) - Min ( \frac{LCM(a,b)}{HCF(a,b)} )

Please note:  Fraction is Max when --> Num is max & Den is min 
  Fraction is Min when --> Num is Min & Den is max

Max ( \frac{LCM(a,b)}{HCF(a,b)} ) = a*b/1
Min ( \frac{LCM(a,b)}{HCF(a,b)} ) =  \frac{b}{a} 
Difference = ab-  \frac{b}{a}  =  \frac{b}{a}  ( a^2  - 1) =  \frac{b}{a}  (a+1) (a-1)

Kinshook · Answer

Asked: If a and b are two distinct positive integers such that b > a, which of the following expressions is equivalent to the difference between the maximum and the minimum possible values of the ratio of the least common multiple of a and b to the highest common factor of a and b ?

HCF_{min}(a,b) = 1 
 LCM_{max}(a,b)= ab 
 HCF_{max}(a,b) = a 
 LCM_{min}(a,b)= b

Max(\frac{LCM(a,b)}{HCF(a,b)}) = \frac{LCM_{max}(a,b) }{ HCF_{min}(a,b)} = \frac{ab }{ 1} = ab 
 Min (\frac{LCM(a,b)}{HCF(a,b)}) = \frac{LCM_{min}(a,b) }{ HCF_{max}(a,b)} = \frac{b}{a}  
Difference between the maximum and the minimum possible values of the ratio of the least common multiple of a and b to the highest common factor of a and b =  ab - \frac{b}{a} = \frac{b}{a} (a^2 - 1) = \frac{b}{a} (a+1)(a-1)

IMO C

TestPrepUnlimited · Answer

We should try to focus on finding the ratio of LCM / HCF of exotic cases first. Think about the extreme cases.

Let HCF to be 1 so that  a  and  b  have no factors in common but 1. In that case, the LCM is  a*b . Then the ratio is  ab/1 = ab .

Let's also think about the extreme case where  b  is a multiple of  a . Then the HCF would be  a , LCM would be  b  since  b  is already a multiple of  a . Then the ratio would be  b/a .

Now that we listed the two extreme cases, we can see the first case is a maximum and the second would correspond to the minimum.

The difference is now  ab - b/a = \frac{b}{a}*(a^2 - 1) = \frac{b}{a}*(a-1)(a+1)

Ans: C

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If a and b are two distinct positive integers such that b > a, which

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