# Answer to Riddle #8: Remainders

8. What is the smallest positive integer that leaves a remainder of 1 when divided by 2, remainder of 2 when divided by 3, a remainder of 3 when divided by 4, and so on up to a remainder of 9 when divided by 10?

The answer to this, (along with #9 & #16,) was sent to me by Dave Blackston who visited me from fark.com (then corrected by Daya Chand)...

Before reading the answer can I interest you in a clue?

The key to this is to realise that if a number N (eg 104) is to leave a remainder, say 4 when divided by 5 then N + 1 (ie 105) is exactly divisible by 5, similarly if a number (eg 87) is to leave a remainder 7 when divided by 8 then N + 1 (ie 88) is exactly divisible by 8:-

Therefore in our question N + 1 is divisible by 2, 3, 4, 5, 6, 7, 8, 9 & 10

There is a tool here you can use to try different values of N

Meaning N+1 is 2000

N+1 has the following factors:

2✓ 3✗ 4✓ 5✓ 6✗ 7✗ 8✓ 9✗ 10✓ All Factors✗

The lowest value for N + 1 is 2520, which means the lowest value for N is:-

N + 1 must be a multiple of 2

N + 1 must be a multiple of 4 but if it is a multiple of 2 & 4 it is necessarily a multiple of 8

N + 1 must be a multiple of 5 but if it is a multiple of 2 & 5 it is necessarily a multiple of 10

N + 1 must be a multiple of 6 but if it is a multiple of 2 & 9 (18) it is necessarily a multiple of 6

N + 1 must be a multiple of 7

N + 1 must be a multiple of 9

Hence 2 x 4 x 5 x 7 x 9 =

Mathematically the process we are dealing with here is known as lowest or least common multiple.

The key to this is to realise that if a number N (eg 104) is to leave a remainder, say 4 when divided by 5 then N + 1 (ie 105) is exactly divisible by 5, similarly if a number (eg 87) is to leave a remainder 7 when divided by 8 then N + 1 (ie 88) is exactly divisible by 8:-

Therefore in our question N + 1 is divisible by 2, 3, 4, 5, 6, 7, 8, 9 & 10

There is a tool here you can use to try different values of N

Meaning N+1 is 2000

N+1 has the following factors:

2✓ 3✗ 4✓ 5✓ 6✗ 7✗ 8✓ 9✗ 10✓ All Factors✗

The lowest value for N + 1 is 2520, which means the lowest value for N is:-

**2519**

## How do we arrive at N + 1 = 2520?

As stated N + 1 is divisible by 2, 3, 4, 5, 6, 7, 8, 9 & 10 so clearly one solution would be 10! (ie 10*9*8*7*6*5*4*3*2 = 3628800 for N +1) but this is not the lowest possible. Follow the logic below...N + 1 must be a multiple of 2

N + 1 must be a multiple of 4 but if it is a multiple of 2 & 4 it is necessarily a multiple of 8

N + 1 must be a multiple of 5 but if it is a multiple of 2 & 5 it is necessarily a multiple of 10

N + 1 must be a multiple of 6 but if it is a multiple of 2 & 9 (18) it is necessarily a multiple of 6

N + 1 must be a multiple of 7

N + 1 must be a multiple of 9

Hence 2 x 4 x 5 x 7 x 9 =

__2520__is a multiple of 2, 3, 4, 5, 6, 7, 8, 9 & 10Mathematically the process we are dealing with here is known as lowest or least common multiple.

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