the linear congruence 456z 20 (mod n) has no solution then value of n can be

Ob3

Oc4

d5

Clear my choice

To solve this, we need to determine under what conditions the linear congruence \( 456z \equiv 20 \pmod{n} \) has no solution.

### Step 1: Check for solvability
A linear congruence of the form \( az \equiv b \pmod{n} \) has a solution if and only if the greatest common divisor (GCD) of \( a \) and \( n \) divides \( b \). In this case, the congruence is \( 456z \equiv 20 \pmod{n} \). Thus, the GCD of \( 456 \) and \( n \) must divide \( 20 \) for the congruence to have a solution.

### Step 2: Calculate the GCD of \( 456 \) and \( 20 \)
The GCD of \( 456 \) and \( 20 \) is \( \gcd(456, 20) \).

First, let's perform the division:
- \( 456 \div 20 = 22 \) remainder \( 16 \).
- Now, \( \gcd(20, 16) \) since \( 456 \mod 20 = 16 \).
- \( 20 \div 16 = 1 \) remainder \( 4 \).
- Finally, \( \gcd(16, 4) = 4 \).

Thus, \( \gcd(456, 20) = 4 \).

### Step 3: Determine the values of \( n \)
For the linear congruence \( 456z \equiv 20 \pmod{n} \) to **not** have a solution, the GCD of \( 456 \) and \( n \) must not divide \( 20 \). Since \( \gcd(456, 20) = 4 \), if \( n \) has a GCD with 456 that is a divisor of 4 and does **not** divide 20, the congruence will have no solution.

Given the options:

- \( \mathbf{n = 3} \): The GCD of 456 and 3 is 3, which does not divide 20.
- \( n = 4 \): The GCD of 456 and 4 is 4, which divides 20.
- \( n = 5 \): The GCD of 456 and 5 is 1, which divides 20.

### Conclusion
The linear congruence \( 456z \equiv 20 \pmod{n} \) has no solution when \( \mathbf{n = 3} \).

So, the correct choice is **Option b (3)**.

Would you like more details or have any questions?

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Here are some related questions you might find interesting:

1. What is the general method to solve linear congruences of the form \( az \equiv b \pmod{n} \)?
2. How do you find the GCD of two numbers using the Euclidean algorithm?
3. How can you determine if a linear congruence has multiple solutions?
4. How would the solution process change if the modulus \( n \) were a prime number?
5. Can a linear congruence have infinitely many solutions?

**Tip:** Always verify if the GCD condition is satisfied when checking the solvability of a linear congruence.

Learn how to determine the value of n such that the linear congruence 456z ≡ 20 (mod n) has no solution. This problem involves modular arithmetic and the application of the Chinese Remainder Theorem. Suitable for advanced high school students and beyond.

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