Summing consecutive numbers

Number Theory Level 1

The sum of any 3 consecutive positive integers must be divisible by 3.
The sum of any 5 consecutive positive integers must be divisible by 5.
The sum of any 7 consecutive positive integers must be divisible by 7.

Is it true that the sum of any 9 consecutive positive integers must be divisible by 9?

Yes, it is true No, it is not true

Pi Han Goh by Pi Han Goh , 30, Malaysia

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2 solutions

Let the 9 consecutive numbers be n , n + 1 , … , n + 8 n,n+1,\ldots,n+8 where n is a positive integer.

The sum of the 9 consecutive numbers is 9 n + 36 = 9 ( n + 4 ) 9n+36=9(n+4) which is divisible by 9.

2 Helpful 0 Interesting 0 Brilliant 0 Confused

1+2+3+4+5+6+7+8+9 = 45

2+3+4+5+6+7+8+9+10 = 54

they are both divisible by 9, so the answer is Yes

0 Helpful 0 Interesting 0 Brilliant 0 Confused

This is wrong.

I'm asking whether it is true for any 9 consecutive positive integers. You have only shown me that it works for 2 sets of 9 integers.

Pi Han Goh - 4 years, 6 months ago

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