Number System: Factors and Coprimes


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