what is the smallest number with factors 1,2,3,4,5,6,7,8,9 and 10?

Mathematics
Tutor: None Selected Time limit: 1 Day

What is the smallest number you can have with all factors 1-10?

Apr 4th, 2015

We can ignore 1, since any counting number is divisible by 1.
So our prime factor each of the counting numbers from 2 to 10 
2 = 2
3 = 3
4 = 2*2
5 = 5
6 = 2*3
7 = 7
8 = 2*2*2
9 = 3*3 
10 = 2*5 
The LCM of all those must have as many factors of
each prime that appears in any factorization 
2 appears at most 3 times as a factor of 8
3 appears at most 2 times as a factor of 9
5 appears at most 1 time as a factor if 5 and 10
7 appears at most 1 time as a factor of 7
So the LCM has 
3 factors of 2, 2 factors of 3, and 1 facor each of 5 and 7
So LCM=2*2*2*3*3*5*7=2520

Apr 4th, 2015

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