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

2The 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 less than that of a number $N$ = $\phi (N)$$ = 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 = ${N^{\left( {\displaystyle\frac{\text{Number of factors}}{2}} \right)}}$ = ${N^{\left( {\displaystyle\frac{{(p + 1).(q + 1).(r + 1)....}}{2}} \right)}}$