Definition 1. An integer n is even if n=2k for some integer k. Definition 2 . An

Question

Definition 1. An integer n is even if n=2k for some integer k.

Definition 2. An integer n is odd if n= 2k+1 for some integfer k.

Proposition 3. The sum of two consecuitive integers is odd.

Explanation / Answer

3. sum of two consecutive integers In two consecuitive integers one should be even and the other odd. so in general two consecutive integers can be represented as 2k and 2k+1. their sum is given as sum = 2k+2k+1 = 4k+1 => 4k + 1 can be written as 2(2k) +1 let 2k = K1 ; since k is an integer 2k is also an integer => K1 is an integer =>4k+1 = 2K1 +1 which is of the form of a odd number so sum of two consecutive integers is a odd number. 4. n is even. so n can be represented as n = 2k. Now n^2 = (2k)^2 = 4k^2 = 2(2k^2) let 2k^2 = K2 ; since k is an integer 2k^2 is also an integer => K1 is an integer so n^2 = 2K2 which is of the form of an even integer so if n is even n^2 is also even

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