00:01
Okay, we're gonna do some congruences.
00:05
So first off is this one.
00:07
So we can divide both sides by eight.
00:11
So we get x is the same as three x mod seven, congruent to.
00:17
So the only solution to that equation is x equals, it's actually x equals zero.
00:31
How'd they get a one in there? the next one, we'll come back to the first one.
00:37
You get to subtract a one from both sides.
00:40
And then 118 is six mod seven, because 112 is a multiple, excuse me, mod eight.
00:53
Because 112 is a multiple of eight.
00:56
So this seven x has to be six mod eight.
01:02
So x equals two, right? cause seven times two is 14, which is six modulo eight.
01:16
The next one gives us 12 x is 15 mod one.
01:24
But modulo one, everything is divisible by one.
01:29
So everything is zero mod one.
01:33
So x can be anything at all.
01:37
In part b, in fact, it's anything.
01:41
We can take two and then add multiples of eight to it or subtract them.
01:47
The first one is x equals zero, seven, 14, so on.
01:54
So there'll be zero mod seven anyway.
01:57
And nothing else works up there.
02:04
We wanna solve this congruence modulo seven.
02:09
So we wanna think about the possible remainders mod seven, which are one through six.
02:25
So if we take the first six powers, let's just take a look at this.
02:30
We take this first six powers, we get all the numbers one through six.
02:38
And when we get up to six, it's one, okay? so that means any multiple of six is still gonna be one, cause we're just gonna keep repeating the same pattern over and over.
02:57
So that means by the time we get to 30, it's gonna be one...