00:01
Okay, so we have these equivalence relations working in modular arithmetic, and we want to find representations which are in a certain range.
00:11
So the first part, we want to find some integer a, which is equal to 17, mod 29, but we want a to be between minus 14 and 14.
00:22
So the point is with each of these questions, if we add or subtract a multiple of the number that we're working, modulo with respect to, then this is the same as adding or subtracting as zero.
00:37
So 17 is the same as 17 minus 29 when we're working mar 29, right? because 29, mar 29 is 0, so all we're doing is subtracting 0 here.
00:52
Well, 17 minus 29.
00:55
This is just minus 12.
00:59
So this is minus 12.
01:02
Mod 29 so we've already found our a if we take a to be minus 12 here then we've shown that a is equal to 17 mod 29 so all we did was subtract 29 here and this gives us an equivalent expression now we're going to do basically exactly the same for parts 2 and 3 we want to find a such that a is equal to 31 mod 10 so all we're going to do is add a multiple of 10 so 31 mod 10 well this is equal to 31 plus any multiple of 10 so now i'm going to write 10 times 5 mod 10 so this mod 10 and this should have been like this sorry mod 10 so any multiple of 10 is obviously zero mod 10.
02:02
So this is the same as adding 0, but it's also the same as adding 50.
02:08
So this is 81, mod 10.
02:13
So we find our number a, we're going to take a to be equal to 81.
02:18
And then a is equal to 31 mod 10...