I just found a number with an interesting property:

When I divide it by 2, the remainder is 1.

When I divide it by 3, the remainder is 2.

When I divide it by 4, the remainder is 3.

When I divide it by 5, the remainder is 4.

When I divide it by 6, the remainder is 5.

When I divide it by 7, the remainder is 6.

When I divide it by 8, the remainder is 7.

When I divide it by 9, the remainder is 8.

When I divide it by 10, the remainder is 9.

It's not a small number, but it's not really big, either.

Find the smallest number with such property.

When I divide it by 2, the remainder is 1.

When I divide it by 3, the remainder is 2.

When I divide it by 4, the remainder is 3.

When I divide it by 5, the remainder is 4.

When I divide it by 6, the remainder is 5.

When I divide it by 7, the remainder is 6.

When I divide it by 8, the remainder is 7.

When I divide it by 9, the remainder is 8.

When I divide it by 10, the remainder is 9.

It's not a small number, but it's not really big, either.

Find the smallest number with such property.

## 5 comments:

Remainder on dividing by any K (K from 2 to 10) is K -1. So the number can be lcm(1,2,3,...10)-1 = 2519

119

I believe the solution is 2,519.

From the first statement, having a remainder of 1 means it is one less than an number divisible by 2.

From the second statement, having a remainder of 2 means it is one less than a number divisible by 3.

From the third statement, having a remainder of 3 means it is one less than an number divisible by 4.

..etc... for the rest of the statements.

This means that it is one less than a number that is divisible by 2,3,4,5,6,7,8,9,10. That means we need the Least-Common-Multiple (LCM) of those nine numbers.

Getting the prime factorization of each of those numbers.

2 = 2^1

3 = 3^1

4 = 2^2

5 = 5^1

6 = 2^1 * 3^1

7 = 7^1

8 = 2^3

9 = 3^2

10 = 2^1 * 5^1

The LCM is the product of the highest exponents for each of those bases.

So the LCM in this case = 2^3 * 3^2 * 5^1 * 7^1 = 8*9*5*7 = 2,520.

Our answer is one less than that number, so our answer is 2,519.

You can do the division with each of those to check that the remainders are correct.

I go with Josh D's answer and explanation : - 2519.

Yes, 2519

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