Expert: Josh - 2/21/2005

Question
what is the smallest whole number which is exactly divisable by all the numbers from 1 to 10 inclusive?

Answer
Hi Harry,

Write down all the prime numbers in the range from 1 to 10.
They are include P={2,3,5,7}.
At the very least, the number that you are looking for cannot be smaller than the product of all these prime numbers (this follows from the prime factorization theorem).
The remaining numbers are
4=2*2=2^2,
6=2*3,
8=2*2*2=2^3,
9=3*3=3^2 and
10=2*5.

Next, observe the prime factors that are missing from Q=2*3*5*7 (including repeating prime numbers).
For example, 4=2*2, but we only have one lot of 2 in Q.
So, expand Q and let Q=2*2*3*5*7.
Then, consider 6=2*3. Nothing to be done. As 2 and 3 are already contained in Q.
Then, consider 8=2*2*2, we already have two lots of 2 in Q, so we need one more.
Expand Q and let Q=2*2*2*3*5*7.
Finally, consider 9=3*3. We only have one 3 in Q at the moment. So, expand Q and let Q=2*2*2*3*3*5*7.
This is what you want.

I hope you get the general idea. I have not bother checking the answer as I'm pressed for time.

Cheers.