Each positive integer is either a prime or the product of powers of primes.
Moreover, in the latter case, if an integer is written as a product of powers of primes in ascending order, then no other such factorization is possible, i.e., factorization is unique.
Example:
540 = 5 × 108 = 5 × [6 × 18] = 5 × [(2 × 3) × (2 × 3 × 3)]
= $2^2 \times 3^2 \times 5^1$
The above factorization is unique.
Two numbers are relatively prime if their factorizations have no common factors.
Example: Are 540 and 539 relatively prime?
Factorization of 540 =$\space2^2 \times 3^2 \times 5^1$ Factorization of 539 = $7^2 \times 11^1$
Since they do not have any common factors, they are relatively prime.
Number (a) has multiplicative inverse mod m if and only if a and m are relatively prime.
Stated in other words:
Number (a) has multiplicative inverse mod m if and only if gcd(a,m) = 1 Where 𝑚 ≥ 2 and 1 ≤ 𝑎 < 𝑚
Factorization of 26 = $2^1 \times 13^1$
Therefore, any number that has a factor of 2 or 13 will not have a multiplicative inverse in mod 26.