I am thinking of a positive number, less than 500, that is divisible by every number from 1 to 10 (inclusive), with one exception. What numb

I am thinking of a positive number, less than 500, that is divisible by every number from 1 to 10 (inclusive), with one exception. What number am I thinking of? Prove that the answer is unique.

Mmm

well if it is divisable by 10 then it is divisable by 2 and 5 as well      10,2,5

If it is divisavle by 9 then it is divisable by 3                                     9,3

If it is divisable by 8 then it is divisable by 4 and 2 as well                8,4,2

If it is divisable by 6 then it is divisable by 3 and 2 as well

All numbers are diviable by 1

So I am probably looking for a number that is not divisable by 7 but is divisable by all the others

Lets see

10*9*8*6=4320     Mmm     that is a bit bigger than 500    

The lowest common multiple of 6 and 9 is 18

The lowest common multiple of 8 and 10 is 40

18*40=720      Still too big

18=2*3*3       and 40 = 2*2*2*5

So the lowest common multiple of 18 and 40 is  2*2*2*5*3*3 = 360

So I do believe that the number you seek is 360.

Now 360 is the smallest common multiple of 1,234,5,6,8,9 and 10  

But it is not divisable by 7.

I am convinced that this is the only number that meets all these criteria but I don't know how to set out a proof 

With 7 out we have:   5*8*9=360    (divisible by: 1,2,3,4,5,6,8,9,10)

with 7 in we have:    5*7*8*9 =2520   (divisible by:  1,2,3,4,5,6,7,8,9,10)