Math Problem Statement

For which of the following 3-tuples of integers (a,b,c) do we have a solution to the system of simultaneous linear congruences: n≡a(mod4) , n≡b(mod7) , n≡c(mod28) ?

(1,6,6)

(0,1,8)

(1,1,9)

(1,2,14)

(2,4,18)

(1,2,23)

Solution

To determine which of the given 3-tuples (a,b,c)(a, b, c) have a solution to the system of simultaneous linear congruences:

na(mod4)n \equiv a \,(\text{mod}\, 4) nb(mod7)n \equiv b \,(\text{mod}\, 7) nc(mod28)n \equiv c \,(\text{mod}\, 28)

we need to check whether a value of nn exists that satisfies all three congruences simultaneously for each tuple (a,b,c)(a, b, c).

Relationship Between the Congruences

  • nc(mod28)n \equiv c \,(\text{mod}\, 28) implies nc(mod4)n \equiv c \,(\text{mod}\, 4) and nc(mod7)n \equiv c \,(\text{mod}\, 7).
  • Therefore, for a solution to exist, aa must be congruent to cmod4c \mod 4, and bb must be congruent to cmod7c \mod 7.

Checking Each Tuple

  1. Tuple (1,6,6)(1, 6, 6):

    • c=6c = 6
    • Check 61(mod4)6 \equiv 1 \,(\text{mod}\, 4)? No.
    • Check 66(mod7)6 \equiv 6 \,(\text{mod}\, 7)? Yes.

    No solution exists since cc is not congruent to amod4a \mod 4.

  2. Tuple (0,1,8)(0, 1, 8):

    • c=8c = 8
    • Check 80(mod4)8 \equiv 0 \,(\text{mod}\, 4)? Yes.
    • Check 81(mod7)8 \equiv 1 \,(\text{mod}\, 7)? No.

    No solution exists since cc is not congruent to bmod7b \mod 7.

  3. Tuple (1,1,9)(1, 1, 9):

    • c=9c = 9
    • Check 91(mod4)9 \equiv 1 \,(\text{mod}\, 4)? Yes.
    • Check 91(mod7)9 \equiv 1 \,(\text{mod}\, 7)? Yes.

    A solution exists since both conditions are satisfied.

  4. Tuple (1,2,14)(1, 2, 14):

    • c=14c = 14
    • Check 141(mod4)14 \equiv 1 \,(\text{mod}\, 4)? No.
    • Check 142(mod7)14 \equiv 2 \,(\text{mod}\, 7)? Yes.

    No solution exists since cc is not congruent to amod4a \mod 4.

  5. Tuple (2,4,18)(2, 4, 18):

    • c=18c = 18
    • Check 182(mod4)18 \equiv 2 \,(\text{mod}\, 4)? Yes.
    • Check 184(mod7)18 \equiv 4 \,(\text{mod}\, 7)? Yes.

    A solution exists since both conditions are satisfied.

  6. Tuple (1,2,23)(1, 2, 23):

    • c=23c = 23
    • Check 233(mod4)23 \equiv 3 \,(\text{mod}\, 4)? No.
    • Check 232(mod7)23 \equiv 2 \,(\text{mod}\, 7)? Yes.

    No solution exists since cc is not congruent to amod4a \mod 4.

Conclusion:

The 3-tuples (1,1,9)(1, 1, 9) and (2,4,18)(2, 4, 18) are the ones for which a solution to the system of simultaneous linear congruences exists.

Would you like more details, or do you have any questions?

Here are 5 follow-up questions:

  1. What is the importance of the moduli in determining the consistency of congruences?
  2. How can the Chinese Remainder Theorem help when moduli are not pairwise coprime?
  3. What happens if the moduli are not coprime in a system of linear congruences?
  4. How can systems of linear congruences be applied in cryptography?
  5. What are the steps to check for a solution manually when the moduli are large?

Tip: Always check the congruences modulo smaller factors before dealing with larger composite moduli. This simplifies the problem and makes it easier to spot inconsistencies.

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Math Problem Analysis

Mathematical Concepts

Modular arithmetic
Linear congruences
Chinese Remainder Theorem

Formulas

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Theorems

Chinese Remainder Theorem

Suitable Grade Level

Advanced High School and above