Positive number less than 500 divisible by 1 to 10 with one exception and how to prove the answer.

Question

"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."

I am having specifically trouble with how to prove the uniqueness of the answer which is 360, my so far "trial and error" method feels too messy and not exact to be a proof.

Answer 1

Let the positive number less than 500 is x.

x is divisible by every number from 1 to 10.

= {2,3,4,5,6,7,8,9,10}

Step1

Start testing divisibility from 10.

10 prime factors are 1,2, and 5.So remove 2,5 from above product.

= {3,4,6,7,8,9,10}

Step2

9 prime factors are 3 and 3.So remove 3 from above product. Keep another 3 aside.

= {4,6,7,8,9,10}

Step3

8 prime factors are 2*2*2. And 2 is already removed.

Just modify as 8 factors are 4*2. So remove 4 from above product. Keep another 2 aside.

= {6,7,8,9,10}

Step4

In 2,3 steps we keep 3,2 aside. Just modify as 3*2 = 6.

So remove 6 from above product.

= {7,8,9,10}

Step5

7 prime number. So should keep it.

= {7,8,9,10}

Step6

6,5,4,3,2,1 all are removed.

The remained product =  7*8*9*10

In above product removing of 7 will not impact others. Since 7 is prime number.

Removing of 8,9,10 will impact other removals such as 1,2,3,4,5,6.

So remove 7 from above product.

Find prime factor the numbers of 8 , 9 ,10.

8 = 2 *4= 2 * 2* 2 = 2³

9 = 3*3 = 3²

10 = 2* 5 = 2 * 5

LCM = 2³ * 3² * 5 = 360

Solution is 360.