Two consecutive odd numbers

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Question from Alicia, a student:

The sum of two consecutive odd numbers is divisible by 4.

Can you make similar statements about the sum of three consecutive odd
numbers and of four consecutive odd numbers?
Generalize your findings.

I tried :
n= smallest odd number
therefore the next = n, n+2 and the second =n+4 and the third = n+6
Now n+ n+2+n+4+n+6/4
Please help I am not sure of this I am just trying

Hi Alicia,

Every odd number is one more than an even number so odd number you start with can be written 2k + 1 for some integer k. Thus two consecutive odd numbers can be written

2k + 1 and 2k + 3

Thus their sum is

2k + 1 + 2k + 3 = 4k + 4 = 4(k + 1)

which is divisible by 4.

Penny

Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.