proving even or odd numbers in forms

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prove that any odd number is either of the form 4k+1 or of the from 4k+3, k belongs to Z

Oct 1st, 2015

Thank you for the opportunity to help you with your question!

Let m in Z be odd.

Using division algorithm, there exits unique p and t in Z such that

m=4p+t and 0<=t<=4

Therefore 4p is even, meaning t is not equal to zero and t is not equal to 2.

Otherwise

m=4p or m=4p+2 which are both even.

Therefore we conclude that

m=4p+1 and m=4p+3 are odd.

Please let me know if you need any clarification. I'm always happy to answer your questions.
Oct 1st, 2015

I dont understand why t can't be equal to 0 or 2.

Oct 1st, 2015

I dont understand why t can't be equal to 0 or 2.

Oct 1st, 2015

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