# Is there a formula or way to calculate which numbers we can divide a certain number to only produce non decimal numbers?

#### ak730

##### New member
Hi,

Not sure if Im explaining correctly.

Is there a formula or way to calculate which numbers we can divide a certain number to only produce non decimal numbers?

For example:
466

Only 2 and 233 will produce a non decimal number.

Hi,

Not sure if Im explaining correctly.

Is there a formula or way to calculate which numbers we can divide a certain number to only produce non decimal numbers?

For example:
466

Only 2 and 233 will produce a non decimal number.
Yes, it's called "prime factorization":
[imath]466 = 2 \cdot 233[/imath]

So 466 can only be divided "evenly" by the primes 2 and 233 (ie. without any extra decimals.)

A more illuminating example would be [imath]180 = 2^2 \cdot 3^2 \cdot 5[/imath]. 180 can be divided evenly by any combination of two 2's, two 3's and a 5:
[imath]2, 3, 2^2, 5, 2 \cdot 3, 3^2, 2 \cdot 5, 2^2 \cdot 3, 3 \cdot 5, 2 \cdot 3^2, 2^2 \cdot 5, 2 \cdot 3 \cdot 5, 2^2 \cdot 3^2, 3^2 \cdot 5, 2 \cdot 3^2 \cdot 5, 2^2 \cdot 3^2 \cdot 5[/imath]

or 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180.

-Dan

Hi,

Not sure if Im explaining correctly.

Is there a formula or way to calculate which numbers we can divide a certain number to only produce non decimal numbers?

For example:
466

Only 2 and 233 will produce a non decimal number.
What you're asking for is the "factors" of 466. These are its divisors, numbers that evenly divide it, producing an integer.

There is no simple "formula", but once you have found the prime factors (which can take some time), @topsquark's suggestion finds all factors by an orderly process. Can you see what that process is?

Now that you have words for what you want, you should be able to search for fuller explanations of these ideas.

Not sure if Im explaining correctly.Is there a formula or way to calculate which numbers we can divide a certain number to only produce non decimal numbers?
For example: 466 Only 2 and 233 will produce a non decimal number.
The easiest way to answer is by way of example.
Step one is to find the prime factors. [imath]19,132,858,800=2^4*3^3*5^2*11^6[/imath]
Step two is to list each exponent plus one: [imath](4+1),(3+1),(2+1),(6+1)[/imath]
Step three find the product of those: [imath](5)\cdot(4)\cdot(3)\cdot(7)=420[/imath]
That tell us that there are [imath]420[/imath] positive integral factors of [imath]19,132,858,800[/imath]
If you work out why that works you will understand your own question.
[imath][/imath][imath][/imath]

Hi,

Not sure if Im explaining correctly.

Is there a formula or way to calculate which numbers we can divide a certain number to only produce non decimal numbers?

For example:
466

Only 2 and 233 will produce a non decimal number.
For the record 466/1 and 466/466 will also be non decimal numbers.

For the record 466/1 and 466/466 will also be non decimal numbers.
466/1 = 466.0000000

466/466 = 1.0000000

I think there is a confusion about the definition of "non-decimal number". A better classification would be non-recurring decimal - as in π) or recurring decimal (as in 4/33 = 0.121212...)

Last edited:
I think there is a confusion about the definition of "non-decimal number". A better classification would be non-recurring decimal - as in π) or recurring decimal (as in 4/33 = 0.121212...)
I think that it is reasonable to assume that divisor means an integer, (SEE HERE).
Of course I am fully aware that mathematical definitions are not ever set in stone.
It is clear that [imath]\dfrac{4}{0.2}=20[/imath] or [imath]\dfrac{4}{\frac{2}{10}}=20[/imath] but very few would say that [imath]0.2[/imath] is a divisor of [imath]4[/imath].
But reply #6 is apt.
[imath][/imath][imath][/imath]

[imath][/imath][imath][/imath][imath][/imath]