Chapter 1 (Revision): LCM & HCF: Someone asked...

The HCF and LCM of two numbers are 12 and 5040 respectively. If one of the number is 420, what is the other number?
If the 2 numbers are A and B, rewriting what we are given:

• HCF of A and B = 12 (which is the same as 2 x 2 x 3)
• LCM of A and B = 5040

Now, to find HCF of A and B, we usually do prime factorisation of each number, then 'circle' the common factors.
In other words, with prime factorisation,

• A = 2 x 2 x 3 x ? x ? x ... x ?
• B = 2 x 2 x 3 x ? x ? x ... x ?
• {where we do not know what are the factors and how many of them at this point}

Now, look at LCM... if we were to use long division method to find the LCM, it means....

• 2 ) A, B
• 2 ) A', B'
• 3 ) A'', B''
• ? ) A''', B''
• ................

So, it means LCM of these 2 numbers A and B = 2 x 2 x 3 x ? x ... x ? (but we have to bear in mind that 2, 2 and 3 are common for both numbers)
Given that one of the numbers is 420, we would be able to find the factors that this number contributes to the LCM.

• Prime Factorisation of 420 = 2 x 2 x 3 x 5 x 7
• Now, we know that LCM of A and B, 5040 is equal to 2 x 2 x 3 x 5 x 7 x ?
• The remaining factor: 5040 ÷ (2 x 2 x 3 x 5 x 7) = 12
• Hence the other number is a product of 2 x 3 x 3 x 12 = 144