Is there an integer [math: z] such that [math: 504z≡7(mod385)] and [math: 0≤z

Question

Is there an integer [math: z] such that [math: 504z≡7(mod385)] and [math: 0≤z<385]? If there is, enter such an integer [math: z]. If there isn't, enter NA.

BlackTom AI · Accepted Answer

gcd(504,385) = 7, so a solution exists because 7 divides the right-hand side 7.

1) Relevant concepts:
- A linear congruence a z ≡ b (mod m) has a solution iff d = gcd(a,m) divides b.
- If d | b, divide the congruence by d to get a reduced congruence with coprime coefficient and modulus, then find the modular inverse to solve for z.

2) Complete reasoning and intermediate calculations:
- Compute gcd(504,385):
  504 − 385 = 119
  385 − 3·119 = 28
  119 − 4·28 = 7
  28 − 4·7 = 0
  So gcd(504,385) = 7.
- Since 7 | 7, divide the congruence 504 z ≡ 7 (mod 385) by 7:
  (504/7) z ≡ 1 (mod 385/7) → 72 z ≡ 1 (mod 55).
- Reduce 72 mod 55: 72 ≡ 17 (mod 55). So 17 z ≡ 1 (mod 55).
- Find inverse of 17 modulo 55 using the extended Euclidean algorithm:
  55 = 3·17 + 4
  17 = 4·4 + 1
  Back-substitute: 1 = 17 − 4·4 = 17 − 4·(55 − 3·17) = 13·17 − 4·55.
  Thus 13·17 ≡ 1 (mod 55), so the inverse of 17 is 13.
- Multiply both sides of 17 z ≡ 1 (mod 55) by 13 to get z ≡ 13 (mod 55).

3) All solutions in the original range:
- The general solution is z = 13 + 55t for integer t. Values with 0 ≤ z < 385 correspond to t = 0,1,...,6, giving z = 13, 68, 123, 178, 233, 288, 343. Any of these satisfy the original congruence.

4) Why the correct answer is correct:
- We reduced the original congruence by gcd and solved the reduced congruence using the modular inverse; z = 13 is a valid representative in the required range and indeed satisfies 504·13 ≡ 7 (mod 385).

5) Other options: Not applicable (short-answer). If other numbers were proposed, only the seven listed above are valid in 0 ≤ z < 385; any other number would not satisfy the congruence.

Conclusion: The integer z = 13 (one valid representative) matches the given correct answer.

Is there an integer [math: z] such that [math: 504z≡7(mod385)] and [math: 0≤z<385]? If there is, enter such an integer [math: z]. If there isn't, enter NA.简答题

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