Prove that if n is an integer and 3n + 2 is even, then n is...

Prove that if n is an integer and 3n + 2 is even, then n is even using a proof by contraposition. Show that if a | b and b | a where a and b are non-zero integers, then a = b or a = -b. Compute the following via the division algorithm (i.e. n=dq+r with 0 ? r < d). Show your work. 67397 mod 7 -1234 mod 12 If a ? 3 (mod 11) and b ? 9 (mod 11): Compute a - b (mod 11) Compute b^2 (mod 11)

Transcript

00:01 Okay, let's get started with the first statement.

00:06 Let's, we are going to prove that 3n plus 2 is even.

00:12 If 3n plus 2 is even, then n is even.

00:16 Now, let's suppose by contradiction that n is odd.

00:28 We are going to show that 3n plus 2 is going to be odd.

00:32 So we are going to find a contradiction.

00:35 Okay, now 3n plus 2 can be written as n plus 2 n plus 2 n plus which in turn can be written as n plus 1 plus 2 n plus 2.

01:02 Now this one is clearly an odd number and now if n is odd this one is even.

01:20 In particular an even number plus an odd number is going to give us an odd number and this is clearly a contradiction.

01:33 So n must be even.

01:39 Okay, now let's take a look at the second one.

01:44 If a divides b, this implies that b can be written as ka with k belonging to the set of integers.

02:04 Similarly, if b divides a, then a can be written as a can be written as hb with h another integer.

02:18 What do we have here? we have that b is equal to k multiplied by hb, which is k multiplied by h multiplied by b.

02:33 So what do we have here? we have that b is equal to this guy, so k multiplied by h is equal to 1.

02:41 But the only two integers such that k and the only two integers k and h such that k multiplied by h is equal to one are given by plus or minus one so in particular what do we have if k and h are equals to plus or minus one this thing means that a is equal to plus or minus b now let's go on here we need to use the division algorithm the division algorithm so let's divide 70 67 39 7 so it's 67 39 7 let's divide it by 7 okay so what are we going to do first of all we can multiply 7 by 7 7 by 9 and we are going to get 63...

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