00:01
Hello, we have to solve the modular equation 17x is congruent to 18 modulo 29.
00:10
First note that 17 and 29 are co -prime.
00:15
So, there exists a modular inverse of 17 modulo 29.
00:22
To find the modular inverse, we apply the extended euclidean algorithm on 17 and 29 to get minus 7 times 29 plus 12 times 17 is equal to 1.
00:35
Taking modulo 29 of both sides, we get 12 times 17 because this term drops out.
00:45
12 times 17 is congruent to 1 modulo 29.
00:48
So, 12 is the modular inverse of 17 modulo 29.
00:52
Now, in this equation, if you multiply 12 to both sides of this equation, we get 12 times 17x is congruent to 12 times 18 modulo 29.
01:05
But 12 times 17 is congruent to 1 modulo 29.
01:10
So, it is 1 times x on the left hand side and 12 times 18 is 216.
01:17
So, it's 216 on the right hand side.
01:20
But 216 is congruent to 13 modulo 29.
01:24
So, we have that x is congruent to 13 modulo 29 and this is the unique solution of this modular equation.
01:34
In part b, we have to solve the modular equation 200x is congruent to 1 modulo 1001.
01:48
Now, if we apply the euclidean algorithm on 1001 and 200, extended euclidean algorithm will get 1001 minus 5 times 200 is equal to 1.
02:02
And so, if we take modulus 1001 of both sides, we'll get this term will drop out.
02:09
Minus 5 times 200 is congruent to 1 modulo 1001.
02:15
So, minus 5 is the modular inverse of 200 modulo 1001.
02:21
Note one thing, because this came out to be 1, 200 and 1001 are co -prime.
02:27
So, there will exist a modular inverse of 200 modulo 1001.
02:34
And note that a solution of this equation is a modular inverse of 200 modulo 1001.
02:40
Since the modular inverse is always unique, minus 5 is the unique is equal to minus 5, which is the modular inverse of 200, is the unique solution of this equation modulo 1001.
02:55
Generally, when we give a number modulo 1001, we want to represent it by a number which is positive, that is between 0 in this case, it should be between 0 and 1001.
03:09
To convert this to a number between 0 and 1001, we have, we write minus 5 is congruent to minus 5 plus 1001, which is equal to 96.
03:19
So, minus 5 is congruent to 996 modulus 1001.
03:29
So, x is congruent to 996.
03:31
So, x is congruent to 996 modulo 1001 is the unique solution of this equation.
03:48
Now, let us look at this part c...