**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**