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)

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

\[
n \equiv a \,(\text{mod}\, 4)
\]
\[
n \equiv b \,(\text{mod}\, 7)
\]
\[
n \equiv c \,(\text{mod}\, 28)
\]

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

### Relationship Between the Congruences

- \( n \equiv c \,(\text{mod}\, 28) \) implies \( n \equiv c \,(\text{mod}\, 4) \) and \( n \equiv c \,(\text{mod}\, 7) \).
- Therefore, for a solution to exist, \( a \) must be congruent to \( c \mod 4 \), and \( b \) must be congruent to \( c \mod 7 \).

### Checking Each Tuple

1. **Tuple \((1, 6, 6)\):**
   - \( c = 6 \)
   - Check \( 6 \equiv 1 \,(\text{mod}\, 4) \)? No.
   - Check \( 6 \equiv 6 \,(\text{mod}\, 7) \)? Yes.
   
   **No solution exists** since \( c \) is not congruent to \( a \mod 4 \).

2. **Tuple \((0, 1, 8)\):**
   - \( c = 8 \)
   - Check \( 8 \equiv 0 \,(\text{mod}\, 4) \)? Yes.
   - Check \( 8 \equiv 1 \,(\text{mod}\, 7) \)? No.
   
   **No solution exists** since \( c \) is not congruent to \( b \mod 7 \).

3. **Tuple \((1, 1, 9)\):**
   - \( c = 9 \)
   - Check \( 9 \equiv 1 \,(\text{mod}\, 4) \)? Yes.
   - Check \( 9 \equiv 1 \,(\text{mod}\, 7) \)? Yes.
   
   **A solution exists** since both conditions are satisfied.

4. **Tuple \((1, 2, 14)\):**
   - \( c = 14 \)
   - Check \( 14 \equiv 1 \,(\text{mod}\, 4) \)? No.
   - Check \( 14 \equiv 2 \,(\text{mod}\, 7) \)? Yes.
   
   **No solution exists** since \( c \) is not congruent to \( a \mod 4 \).

5. **Tuple \((2, 4, 18)\):**
   - \( c = 18 \)
   - Check \( 18 \equiv 2 \,(\text{mod}\, 4) \)? Yes.
   - Check \( 18 \equiv 4 \,(\text{mod}\, 7) \)? Yes.
   
   **A solution exists** since both conditions are satisfied.

6. **Tuple \((1, 2, 23)\):**
   - \( c = 23 \)
   - Check \( 23 \equiv 3 \,(\text{mod}\, 4) \)? No.
   - Check \( 23 \equiv 2 \,(\text{mod}\, 7) \)? Yes.
   
   **No solution exists** since \( c \) is not congruent to \( a \mod 4 \).

### Conclusion:

The 3-tuples \((1, 1, 9)\) and \((2, 4, 18)\) 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.

Explore solutions to simultaneous linear congruences modulo 4, 7, and 28 with detailed analysis for given tuples (1, 6, 6), (0, 1, 8), (1, 1, 9), (1, 2, 14), (2, 4, 18), and (1, 2, 23). Learn about modular arithmetic, conditions for congruence, and when solutions exist. Suitable for advanced high school students and beyond.

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