00:01
Hello, so going through here, for number 11, we have that 2x is congruent to 6 mod 4.
00:09
So to simplify the congruence, 6x mod 4 equals 2, so the equation becomes 2x is congruent to 2, mod 4, and then we can divide through by 2, since 2 divides both sides and 4.
00:25
So we get that 2x is congruent to 2.
00:28
Again mod four and then we get that x is congruent to one mod two the general solution then is x is equal to one plus 2k where k is any integer and then for number 12 we have that 22 x is congruent to 5 mod 15 so put that here it's mod 15 okay so we that 22 is include to 7 mod 15.
01:02
So the equation here becomes 7x is congruent to 5.
01:06
Again, mod 15.
01:08
And then we find the modular inverse of 7 mod 15.
01:12
The inverse is 13 since 7 times 13 is congruent to 1 mod 15.
01:17
So multiplying both sides by 13, we get that x is congruent to 5 times 13, mod 15.
01:26
And then x is congruent to 65, mod 15.
01:30
And then simplify 65 mod 15.
01:33
You get that x is congruent to 5, mod 15.
01:36
So the general solution is going to be x is equal to 5 plus 15k, where again, k is an integer.
01:44
For number 13, we get 36x, oops, 36x is congruent to 15.
01:55
This is mod 24.
02:00
So 36 is congruent to 12.
02:03
Mod 24.
02:04
So the equation becomes 12x is congruent to 15.
02:08
We divide through by the greatest common divisor, the gcd of 12 and 24, which is 12.
02:14
So we get that x is congruent to 15 over 12, mod 24 over 12, and get that x is congruent to 3, mod 2.
02:24
So the general solution here is going to be x is equal to 3 plus 2k.
02:29
Again, k is any integer.
02:32
And then for number 14, we have 45x is congruent to 15, mod 24.
02:45
So 45 is congruent to 21, mod 24.
02:49
The equation then becomes 21x is congruent to 15.
02:55
Again, mod 24, dividing through by the gcd of 21 and 24...