+0

# 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

0
721
2
+1766

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.

Mellie  Jun 22, 2015

#1
+91469
+10

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

Melody  Jun 22, 2015
Sort:

#1
+91469
+10

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

Melody  Jun 22, 2015
#2
+5

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)

Guest Jun 22, 2015

### 8 Online Users

We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details