(g) For all even integers n, n^2 is a multiple of 4. Prove...

(g) For all even integers n, n^2 is a multiple of 4. Prove each statement. (a) If x is an integer, then x2+5x-1 is odd (b) If integers x and y have the same parity, then x+y is even. The parity of a number tells whether the number is odd or even. If x and y have the same parity, they are either both even or both odd.

Transcript

00:01 In this question, we want to prove that for all integers n, we have that n squared is a multiple of 4.

00:11 So let's first by defining what does it mean for n to be even.

00:19 An integer n is even if that integer n can be written as two times another integer, say p, where p is some integer.

00:40 Next, let's define what does it mean for n square to be a multiple of 4? n square is a multiple of 4 if n squared can be written as 4 times some integer, q, let's say, where q is some integer.

01:11 So to demonstrate this proof, let's start by squaring an even number and see if it is a multiple of 4 which abides by this formula.

01:23 So let's take n -squared.

01:27 N, we will rewrite it as 2p because we know that it is even.

01:33 We will find that n -squared is equal to 4 p -squared.

01:40 Now let's say that p -squared is equal to q.

01:48 N -squared will thus be equal to 4 times q.

01:51 And this number is a multiple of 4 by definition.

02:06 So we started with n squared and ended up with a multiple of 4.

02:11 So this concludes our proof for the first question.

02:15 We have another two questions to answer, or another two proofs.

02:20 Next, we want to show that if x is an integer, then we have that x squared minus 5x minus 1 is odd.

02:30 And this is going to be true for any integer.

02:36 Let's recall that the integers can be divided or the complete set of integers can be divided in the set of even integers and odd integers.

02:57 So to prove the statement for any integer, we need to first prove it for even integers and then again prove it for odd integers and then we have scoped the complete set of integers.

03:10 So first, let's say that x is even.

03:26 In this case, x can be written as 2 times some integer p.

03:33 We saw this in the previous question, or p is an integer.

03:42 So let's take x and square it, extract 5 minus 1.

03:52 Let's see what this gives us with our definition of x we'll have 2p squared minus 5 times 2p minus 1 this is equal to 2 times 2p squared minus 2 times 5p minus 1 this integer here will always be odd sorry even because it is 2 times some integer we couldn't if we want rewrite 2p squared as q.

04:54 We can do the same thing, or let's write it that way they can more clear actually...

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