2024/09/09　約数の個数について

To find how many divisors a number has, factorize the number A into its prime factors:

A = x^a ✕ y^b ✕ z^c.

Then, by calculating (a + 1)(b + 1)(c + 1), you can determine the number of divisors.

The reason this formula works is based on how divisors are formed from the prime factorization of a number.

Let's take a number ( A = x^a \times y^b \times z^c ), where ( x, y, z ) are prime numbers, and ( a, b, c ) are their respective exponents.

A divisor of ( A ) must also be of the form ( x^p \times y^q \times z^r ), where ( 0 \leq p \leq a ), ( 0 \leq q \leq b ), and ( 0 \leq r \leq c ). This means each prime factor can contribute any power from 0 up to its respective exponent.

For ( x^a ), you can choose ( p ) to be any integer from 0 to ( a ), giving you ( a + 1 ) choices (because it includes all powers from 0 to ( a )).

Similarly, for ( y^b ), there are ( b + 1 ) choices for ( q ), and for ( z^c ), there are ( c + 1 ) choices for ( r ).

Since these choices are independent of each other, the total number of divisors is the product of the number of choices:

(a + 1)(b + 1)(c + 1)

This gives you the total number of divisors of ( A ).

The formula for finding the number of divisors of a number based on its prime factorization is a result of classical number theory, and its discovery cannot be attributed to a single person.

That being said, why is it that any number raised to the power of 0 equals 1? I'm already sleepy today, so I'll leave this to tomorrow's me to figure out.