Two numbers having 1 as the only common factor are called co-prime numbers.
For example, 16 and 35 are co-prime number.
Factors of 16: 1, 2, 4, 8, 16
Factors of 35: 1, 5, 7, 35
Here, 1 is the only common factor.

If a number is divisible by both 2 and 3, then it will be divisible by 6 also.
Divisibility by 2
Here all the numbers, 9638, 9640, 9642, 9648 are divisible by 2 as they end with one of these (0, 2, 4, 6, 8) numbers or simply as they are even numbers.
Divisibility by 3
$\Rightarrow$9 + 6 + 3 + 8 = 26
$\Rightarrow$9 + 6 + 4 + 0 = 19
$\Rightarrow$9 + 6 + 4 + 2 = 21
$\Rightarrow$9 + 6 + 4 + 8 = 27
Here only 21 and 27 are divisible by 3 hence only 9642 and 9648 are divisible by 3.
So, these two numbers are divisible by both 2 and 3 and hence divisible by 6.
Divisibility by 8
A number will be divisible by 8 only if the last three digits are divisible by 8.
The last 3 digits of only 9648, i.e. 648 is divisible by 8.
But last three digits of 9642, i.e. 642 is not divisible by 8.
 9648 is divisible by both 6 and 8.

If two given numbers are divisible by a number, then their sum, difference and product is also divisible by that number. 
For example, the numbers 35 and 25 are both divisible by 5 and 
their
i)difference (35-25 = 10) is also divisible by 5
ii)sum (35+25 = 60) is also divisible by 5.
iii)product$(35\times 25=875)$ is also divisible by 5.

Factors of 75 are 1, 3, 5, 15, 25 and 75.
Factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 30 and 60.
Factors of 210 are 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105 and 210.

∴ Common factors of 75, 60 and 120 are 1, 3, 5 and  15.

Take an example and visualize-

i)9 and 4 are co-prime but none of them are prime.

ii)Now, 14 divides 28, but it does not divide 12 and 16.

iii)4 divides 12 and 16 separately

4 also divides 12+16 = 28

Common multiples of any given set of numbers will always be the multiples of the LCM of the numbers.
Now, LCM of 4, 5 and 11 can be found as follows.
Hence, LCM = $2\times 2\times 5\times 11=220$
The multiples of 220 are 440, 660, 880...
Hence, in the given options, the common multiples of 4, 5 and 11 are 220, 440 and 660.

If a number is divisible by a number 'n', it is also divisible by the factors of 'n'.

Factors of 8 = 1, 2, 4, 8

Any number divisible by 8 will also be divisible by 4.

But vice-versa is not true. For example 20 is divisible by 4 but not divisible by 8

For any two numbers x and y, if x is divisible by y, then x is also divisible by each factor of y.
For example, 12 is divisible by 6, and 2 is a factor of 6. so, 12 is also divisible by 2.

Now, factors of 26 = 1, 2, 13, 26

312 is divisible by 26, hence it is also divisible by 13.

Prime factorization is finding the factors of a number that are all prime.

Eg:- 12 = 2 × 2 × 3

36 = 2 × 2 × 3 × 3 etc.

72 = 2 × 2 × 2 × 3 × 3
105 = 3 × 5 × 7

625 = 5 × 5 × 5 × 5

162 = 2 × 3 × 3 × 3 × 3
Hence, only the factorisation of 105 is correct