Prime factorisation is expressing a number as the product of its prime factors.
60 = 2 x 30
     = 2 x 2 x 15
     = 2 x 2 x 3 x 5
(2, 3 and 5 are prime numbers) 
Hence, the prime factorisation of 60 is 2 x 2 x 3 x 5.

Factors of 45 = 1, 3, 5, 9, 15, 45

Factors of 81 = 1, 3, 9, 27, 81

Factors of 27 = 1, 3, 9, 27
$\therefore$ The common factors of 45, 81 and 27 are 1, 3 and 9.

So, the H.C.F of 45, 81 and 27 = 9.

On simplification of the given expression, we get that
3 × 3 × 11 × 101 = 9999.
We can observe that it is the greatest 4 digit number.
Also, we can see that this is an odd number.

Numbers, which do not have any common factor between them other than one, are called co-prime numbers.
 
For example, 3 and 7 are co-prime numbers. They only have 1 as a common factor.

The smallest number that is divisible by 32, 64, 128 is the LCM of 32, 64, 128.
LCM of 32, 64, 128 =

= 2×2×2×2×2×2×2

=128

When 128 is divided by 32, 64 & 128 there is no remainder. So, to get remainder 8 when divided by 32, 64 & 128, 8 is to be added to 128 i.e., 128+8 = 136 

The capacity of each storage unit should be same as well as minimum. The required minimum capacity should be the LCM of 80 kg, 85 kg & 90 kg.

$2\times 2\times 2\times 2\times 5\times 9\times 17=12240$

LCM of $80,85,90=12240$

It is given that LCM = 12; HCF=2
Product of 2 numbers = HCF × LCM of the respective numbers
$\Rightarrow$ 6 $\times$ the other number = 12 $\times$ 2
$\Rightarrow$ 6 $\times$ the other number =  24
$\Rightarrow$the other number  =  $\frac{24}{6}$  =   $4$
 Hence, the other number = 4

LCM of 18, 24 and 32 is 

So,  LCM = $2\times 2\times 2\times 2\times 2\times 3\times 3$
               = 288

Prime numbers have only two factors 1 and the number itself.
Hence, LCM of two prime numbers is always the product of the two numbers.

For example, LCM of 3 and 5 = 3 × 5 = 15
Similarly, LCM of 7 and 11 = 7 × 11 = 77

Multiples of 90 will be 90, 180, ...... and
Multiples of 1 will be 1, 2, 3...90 and so on.
From the above, we can say that, the least common multiple of 1 and 90 will be 90.
The L.C.M. of 1 and any other number, say 'x' is always 'x'.