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#1 2017-10-08 18:49:41

iamaditya
Member
From: Planet Mars
Registered: 2016-11-15
Posts: 739

Help with a proof


Practice makes a man perfect.
There is no substitute to hard work
All of us do not have equal talents but everybody has equal oppurtunities to build their talents.-APJ Abdul Kalam

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#2 2017-10-08 22:41:18

bob bundy
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Registered: 2010-06-20
Posts: 8,151

Re: Help with a proof

hi iamaditya

When I've got a proof like this I find it helpful to consider an example first.

Let's say a = 48 = 2 x 2 x 2 x 2 x 3
and        b = 56 = 2 x 2 x 2 x 7

The hcf = 2 x 2 x 2

and lcm = 2 x 2 x 2 x 2 x 3 x 7

so hcf x lcm = (2 x 2 x 2) x (2 x 2 x 2 x 2 x 3 x 7) = (2 x 2 x 2 x 2 x 3) x (2 x 2 x 2 x 7) = 48 x 56

The common factors occur in both the hcf and lcm and the not-common factors occur in the lcm.  So the re-arrangement allows us to pick out one set of common factors together with one set of not-common factors for the first number and what is left is the factors for the other.  Here's an attempt to make that rigorous:

Suppose a = hcf x N where N is the not-common factors, and similarly b = hcf x M

Then the lcm = all the common factors once and the not-common factors from both = hcf x N x M

So hcf x lcm = (hcf) x ( hcf x N x M) = (hcf x N) x ( hcf x M) = a x b

Hope that helps,

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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