# What is the number of factors of 420

## numere-prime.ro

If "t" is a divisor of "a", then when dividing into factors of "t", only prime numbers will appear that will also appear when dividing "a" and which at most do not matter exponents also when dividing "a" intervened.

For example, 12 is a divisor of 60:
12 = 2 × 2 × 3 = 22 × 3
60 = 2 × 2 × 3 × 5 = 22 × 3 × 5

If "t" is the common divisor of "a" and "b", then "t" only has prime factors that also intervene in "a" and "b", each factor at the smallest strength.

For example, 12 is the common divisor of 48 and 360. From the division into prime factors:
12 = 22 × 3
48 = 24 × 3
360 = 23 × 32 × 5
It is observed that 48 and 360 contain several common factors: 2, 3, 4, 6, 8, 12, 24. Of which, 24 is the greatest common factor (GCF) of 48 and 360.

If two numbers a and b have no other common divisor than 1, gcd (a, b) = 1, the numbers a and b are prime numbers between each other.

If "a" and "b" are not prime numbers between them, then every common factor of "a" and "b" is a factor of the greatest common factor of "a" and "b" because the greatest common factor is that Product of all prime factors that intervene between "a" and "b" at the lowest power. This procedure can be used to find the greatest common divisor of several numbers, as shown in the example below.
Example for the definition of the GCD:
1260 = 22 × 32
3024 = 24 × 32 × 7
5544 = 23 × 32 × 7 × 11
gcd (1260, 3024, 5544) = 22 × 32 = 252