To solve the system of simultaneous linear congruences:

1. $n \equiv 2 \pmod{5}$
2. $n \equiv 3 \pmod{7}$

We need to check which of the given numbers satisfy both congruences.

Checking the numbers:

1. 87:

   • $87 \mod 5 = 2$ (Satisfies $n \equiv 2 \pmod{5}$)
   • $87 \mod 7 = 3$ (Satisfies $n \equiv 3 \pmod{7}$)
   • 87 is a solution.

2. 122:

   • $122 \mod 5 = 2$ (Satisfies $n \equiv 2 \pmod{5}$)
   • $122 \mod 7 = 3$ (Satisfies $n \equiv 3 \pmod{7}$)
   • 122 is a solution.

3. 147:

   • $147 \mod 5 = 2$ (Satisfies $n \equiv 2 \pmod{5}$)
   • $147 \mod 7 = 0$ (Does not satisfy $n \equiv 3 \pmod{7}$)
   • 147 is not a solution.

4. 191:

   • $191 \mod 5 = 1$ (Does not satisfy $n \equiv 2 \pmod{5}$)
   • $191 \mod 7 = 2$ (Does not satisfy $n \equiv 3 \pmod{7}$)
   • 191 is not a solution.

5. 227:

   • $227 \mod 5 = 2$ (Satisfies $n \equiv 2 \pmod{5}$)
   • $227 \mod 7 = 3$ (Satisfies $n \equiv 3 \pmod{7}$)
   • 227 is a solution.

6. 257:

   • $257 \mod 5 = 2$ (Satisfies $n \equiv 2 \pmod{5}$)
   • $257 \mod 7 = 5$ (Does not satisfy $n \equiv 3 \pmod{7}$)
   • 257 is not a solution.

Solutions:

The numbers that satisfy both congruences are:

• 87
• 122
• 227

Would you like more details or have any other questions?

Related Questions:

1. How can you solve systems of congruences with more than two equations?
2. What methods exist to solve simultaneous congruences if the moduli are not coprime?
3. What is the general solution for $n$ if the moduli are coprime?
4. How does the Chinese Remainder Theorem simplify solving such systems?
5. What are the applications of modular arithmetic in real-world problems?

Tip:

When solving simultaneous congruences, try to reduce the problem to simpler cases by testing the conditions on smaller numbers before checking larger ones.