How many factors does a million have?

1 000 000

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- May 8th 2006, 01:10 PMNatasha1How many factors does a million have?
How many factors does a million have?

1 000 000 - May 8th 2006, 01:52 PMThePerfectHackerQuote:

Originally Posted by**Natasha1**

. It is defined as the number of distinct divisors of . For example,

cuz 3 only got 2 factors 1 and 3.

There is a formula, if

is the prime-factorization of

Then,

Thus, since,

Thus, - May 23rd 2006, 08:08 AMsrulikbdns formula!
:) what is the proof?

- May 23rd 2006, 09:10 AMCaptainBlackQuote:

Originally Posted by**srulikbd**

Then any factor of is of the form:

where , and each number of this

form is a factor of .

So how many 's are there? Well the first exponent can take

any one of values the second can ....

Hence the number of factors is:

RonL - May 23rd 2006, 10:31 AMsrulikbdtnx a lot
easy indeed ;)

- May 23rd 2006, 01:25 PMThePerfectHackerQuote:

Originally Posted by**srulikbd**

This one is more advanced but more elegant ;)

**Theorem:**

If is a number theoretic function which is multiplicative. Then, the function definied as, is also.

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Note that,

Since, is multiplicative since,

whenever we have that,

is mulitplicative.

Since,

You have,

because or multiplicativity.