Question

What is "Relation between H.C.F and L.C.M

Answer 1

If x= (a1)^(b1) * (a2)^(b2) * (a3)^(b3) ..... 
and y = (c1)^(d1) * (c2)^(d2) * (c3)*(d3) .... 
where ai's and bi's are prime factors of that particular number... 
LCM means least common multiple.. which means we have to take common factors exactly the same number of times repeated in common and the remaining factors as such. 
Example :-- for LCM of 6 and 9, 6= 3 × 2 and 9 = 3 × 3. So,we take the common factor 3 exactly once because it came only once. and the remaning 2 and 3. => LCM of 6 and 9 is 3 × 3 × 2 = 18. 
HCF means highest common factor.. which means we have to take only the common prime factors the number of times they are repeated. 
Example :-- for HCF of 6 and 9, 6 = 3 × 2 and 9 = 3 × 3. So, we take the only common factor 3 and that too exactly once. So, HCF of 6 and 9 is 3. 
Now, answer for your question : We consider all the common factors once in LCM and once in HCF. So, the common factors are multiplied twice. Similarly we take non-common factors for LCM. So, when we multiply LCM and HCF, we are multiplying all the prime factors of both numbers. Hence the result. 
Example :- We saw how we got LCM and HCF of 6 and 9 in the above example. So, now if we multiply them, we have LCM= 3-->common × 3 × 2 and HCF = 3-->common. So, LCM × HCF= (3 × 3 × 2) ×( 3 ) = (3) × (3 × 2) × (3) = (3 × 3) × (3 × 2 ) =9 × 6. 

Hope this helps:)

Answer 2

The best way is to find a counter-example. I find the following relation: Given a, b, c are positive integers, abc LCM a,b,c HCF a,b HCF b,c HCF c,a HCF a,b,c …. The numbers are written in prime factors, but not in the usual unique representation with prime factors and indices.