Let a number "N" is written in its prime factorization format. $N = {a^p} \times {b^q} \times {c^r}...$

**Formula 1:**The number of factors of a number N

= \(\left( {p + 1} \right) \times \left( {q + 1} \right) \times \left( {r + 1} \right)...\)

**Formula 2:**The sum of factors of a number N

= \(\dfrac{{{a^{p + 1}} - 1}}{{a - 1}} \times \dfrac{{{b^{q + 1}} - 1}}{{b - 1}} \times \dfrac{{{c^{r + 1}} - 1}}{{c - 1}}...\)

**Formula 3:**The number of ways of writing a number as a product of two numbers.

$\dfrac{1}{2} \times \left[ {(p + 1).(q + 1).(r + 1)...} \right]$ (if the number is not a perfect square)

If the number is a perfect square then two conditions arise:

1. The number of ways of writing a number as a product of two distinct numbers = $\dfrac{1}{2} \times \left[ {(p + 1).(q + 1).(r + 1)... - 1} \right]$

2. The number of ways of writing a number as a product of two numbers and those numbers need not be distinct= $\dfrac{1}{2} \times \left[ {(p + 1).(q + 1).(r + 1)... + 1} \right]$

**Formula 4:**The number of co-primes of a number

N=$\phi (N) = {a^p}.{b^q}.{c^r}...$ can be written as $N \times \left( {1 - \dfrac{1}{a}} \right) \times \left( {1 - \dfrac{1}{b}} \right) \times \left( {1 - \dfrac{1}{c}} \right)...$

**Formula 5:**The sum of co-primes of a number

= $\phi (N) \times \displaystyle\frac{N}{2}$

**Formula 6:**The number of ways of writing a number N as a product of two co-prime numbers

= ${2^{n - 1}}$ where 'n' is the number of prime factors of a number.

**Formula 7:**Product of all the factors

= ${N^{\left( {\displaystyle\frac{\text{Number of factors}}{2}} \right)}}$ = ${N^{\left( {\displaystyle\frac{{(p + 1).(q + 1).(r + 1)....}}{2}} \right)}}$