# Thread: counting numbers

1. ## counting numbers

How do i know if 360 is the smallest counting number divisible by all of 4,6,9,and 10

2. Originally Posted by Pinky&The Brain
How do i know if 360 is the smallest counting number divisible by all of 4,6,9,and 10
It isn't 180 is.

Look at the prime factorisations of 4, 6, 9, and 10, we have:

4=2^2,
6=2*3,
9=3*3,
10=2*5.

So anything divisible by all these numbers must be divisible by:

2^2*3^2*5=180

so 180 is the least of these.

RonL

3. Originally Posted by CaptainBlack
So anything divisible by all these numbers must be divisible by:

2^2*3^2*5=180

so 180 is the least of these.

RonL

I do not understand what you did here.

4. Originally Posted by Pinky&The Brain
I do not understand what you did here.
I took all the prime factors of the numbers in the list with the maximum
multiplicity that they appear with in any one number, and then multiplied
them together to find the required number.

It is clear that as 4=2*2, then 2 must appear at least twice in the
factorisation of the desired number, and as it appears no more times
in any one of the factorisations of the numbers in the list it must appear
exactly twice in the prime decomposition of the desired number.

Same for 9=3*3.

The only prime factor of a number in the list now unaccounted for is 5,
so we must have one of those as well.

RonL

5. Thanks... I got it now!