Is there an integer [math: y] such that [math: 504y≡10(mod385)] and [math: 0≤y

Question

Is there an integer [math: y] such that [math: 504y≡10(mod385)] and [math: 0≤y<385]? If there is, enter such an integer [math: y]. If there isn't, enter NA.

BlackTom AI · Accepted Answer

gcd(504,385) = 7, and 7 does not divide 10, so there is no solution; answer = NA.

1) Relevant concept: A linear congruence a y ≡ b (mod n) has a solution iff d = gcd(a,n) divides b. If d | b, there are exactly d incongruent solutions modulo n; if d ∤ b, no solution exists.

2) Compute the gcd using the Euclidean algorithm (showing intermediate steps):
- 504 = 385·1 + 119
- 385 = 119·3 + 28
- 119 = 28·4 + 7
- 28 = 7·4 + 0
Thus gcd(504,385) = 7.

3) Apply the solvability criterion: we require 7 | 10 for any solution to exist. But 10 = 7·1 + 3, so 7 does not divide 10.

4) Alternate reduction viewpoint: Reduce the coefficient mod 385: 504 ≡ 119 (mod 385), so the congruence is equivalent to 119 y ≡ 10 (mod 385). Since gcd(119,385)=7 (same result), the same divisibility obstruction appears.

5) Why this makes the correct answer NA: Because gcd(504,385) = 7 does not divide the right-hand side 10, the congruence has no integer solution y in any residue class, in particular none with 0 ≤ y < 385.

6) Why other possibilities are incorrect: Any claim of an explicit y would contradict the necessary divisibility condition. If 10 had been a multiple of 7 (e.g., 14), then solutions would exist (in fact 7 distinct residues modulo 385), but here 10 is not such a multiple, so no solutions are possible.

Conclusion: There is no integer y with 0 ≤ y < 385 satisfying 504y ≡ 10 (mod 385). The correct entry is NA.

Is there an integer [math: y] such that [math: 504y≡10(mod385)] and [math: 0≤y<385]? If there is, enter such an integer [math: y]. If there isn't, enter NA.Short answer

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