A find d6993008 6993008 b find a value of n for which the...

A) Find d(6993008), ϕ(6993008), b) Find a value of n for which the following sentence is true for every x relatively prime to 6993008. If a ≡ b mod n, then x^a ≡ x^b mod 6993008.

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00:01 So we'd like to find which of these values is false, of the four, which of these statements is false, rather.

00:09 First one, this whole number mod 5 is equal to 2.

00:12 I'm going to claim this is true.

00:15 We can verify that because 10 mod 5 is 0.

00:19 So we can discount any multiple of 10.

00:22 And the first, all the digits except for the last one of any number is a multiple of 10.

00:29 So we can cancel all of this out, mod 5.

00:33 And this produces 2.

00:34 Is 7 mod 5 equal to? and it is.

00:38 Another way to think of that is that all this yada, yada, yada with 5 instead of 7 there will be 0 mod 5.

00:46 Add 2, we get 2.

00:48 Point is, statement a is true.

00:52 Statement b here, for each natural number a, if a squared mod 4 is 0, so a squared is a multiple of 4, 4k for some k, if you take the square root of this, a is equal to 4 times the square root of k.

01:08 In particular, this has to be an integer, so i'm going to call it n.

01:14 So a is 0 mod 4.

01:19 I had to take the square root of 4, so that's 2.

01:21 A is equal to 2 times n.

01:23 So really, this just means that a is even...