00:01
So we're looking for the constant x which is in the integers such that 5 multiplied by that x gives us a remainder of 3 when we divide that multiplicand by 12.
00:16
So a couple of ways of doing this.
00:20
We can solve this by multiplicative inverses or we could use a number line.
00:27
I'll do the inverses first and then i'll show the number line after.
00:31
So to use the multiplicative inverse, use the fact that if we have 5x is congruent to 3 mod 12, what we want to do to get x is to divide both sides by 5.
00:48
Or in other words, find the inverse of 5 mod 12.
00:55
Now what is the inverse of 5.
00:57
If we think about what the inverse means, we're looking for a number y multiplied by 5 that gives 1 mod 12.
01:11
That's what an inverse means in modulo form.
01:15
So if we look really carefully, sort of think about it, and we get we know that 1 mod 25 is congruent to 1 mod 12, so the inverse of 5 is indeed 5 itself...