a) We want to solve the system of congruences:
x ≡ 1072 (3279)
x ≡ 77 (2303)
First, we find the solutions to the two congruences separately. For the first congruence, we have:
3279 = 5 * 2303 + 674
So we can write:
x ≡ 1072 (3279) ≡ 1072 (5 * 2303 + 674) ≡ 1072 (674) (mod 2303)
We can use the Euclidean algorithm to find the inverse of 674 modulo 2303:
2303 = 3 * 674 + 281
674 = 2 * 281 + 112
281 = 2 * 112 + 57
112 = 2 * 57 - 2
Working backwards, we have:
1 = 3 - 2 * (674 - 2 * (281 - 2 * 112 + 2)) = 7 * 674 - 6 * 2303
So we can multiply both sides of the congruence by 674 and simplify:
x ≡ 1072 (674) (mod 2303)
x ≡ 722 (mod 2303)
Now, we can use the same method to solve the second congruence:
2303 = 29 * 77 + 42
77 = 1 * 42 + 35
42 = 1 * 35 + 7
35 = 5 * 7 + 0
Working backwards, we have:
1 = -1 * 29 + 2 * 7
= -1 * 29 + 2 * (42 - 1 * 35)
= 2 * 42 - 3 * 35
= 2 * 42 - 3 * (77 - 42)
= -3 * 77 + 5 * 42
= -3 * 77 + 5 * (2303 - 29 * 77)
= -152 * 77 + 5 * 2303
So we can multiply both sides of the congruence by 152 and simplify:
x ≡ 77 (152) (mod 2303)
x ≡ 497 (mod 2303)
Now we have two congruences that we can solve using the Chinese Remainder Theorem. We need to find integers a and b such that:
x ≡ a (3279 * 2303)
x ≡ b (674 * 152)
To find a, we can use the formula:
a = (77 * 3279 * 152 + 1072 * 674 * 2303) mod (3279 * 2303)
To find b, we can use the formula:
b = (1072 * 674 * 152 + 497 * 3279 * 2303) mod (674 * 152)
Evaluating these formulas, we get:
a = 2258536
b = 602064
So the solution to the system of congruences is:
x ≡ 2258536 (mod 3279 * 2303)
x ≡ 602064 (mod 674 * 152)
To find the unique solution x between 0 and 3279 * 1072, we can use the formula:
x = a + (b - a) * (3279 * 2303) * (674 * 152)^(-1) mod (3279 * 1072)
where (674 * 152)^(-1) is the inverse of 674 * 152 modulo 3279 * 1072. We can find this inverse using the Euclidean algorithm:
3279 * 1072 = 3 * 674 * 152 + 536064
674 * 152 = 1 * 536064 + 36320
536064 = 14 * 36320 + 4944
36320 = 7 * 4944 + 272
4944 = 18 * 272 + 240
272 = 1 * 240 + 32
240 = 7 * 32 + 16
32 = 2 * 16 + 0
Working backwards, we have:
1 = 2 - 1 * 1
= 2 - 1 * (32 - 2 * 16)
= -1 * 32 + 3 * 16
= -1 * 32 + 3 * (240 - 7 * 32)
= 22 * 32 - 3 * 240
= 22 * (272 - 240) - 3 * 240
= -25 * 240 + 22 * 272
= -25 * (4944 - 18 * 272) + 22 * 272
= 472 *