Question

Find the value of \( k \) such that \( 5 \cdot 2^n - 48n + k \) is divisible by 24.

Hint:

When dealing with divisibility problems, use modular arithmetic to check for each value of \( n \), and substitute values into the expression to determine the value of \( k \) that satisfies the condition.

Answer 1

We are asked to find \( k \) such that the expression \( 5 \cdot 2^n - 48n + k \) is divisible by 24. In other words, we need to find \( k \) such that: \[ 5 \cdot 2^n - 48n + k \equiv 0 \pmod{24} \]

Step 1: Analyze the expression

We want the expression \( 5 \cdot 2^n - 48n + k \) to be divisible by 24 for any integer \( n \). To find \( k \), let's start by evaluating the expression for a few values of \( n \).

Step 2: Check for \( n = 0 \)

Substitute \( n = 0 \) into the expression: \[ 5 \cdot 2^0 - 48(0) + k = 5 + k \] For divisibility by 24: \[ 5 + k \equiv 0 \pmod{24} \] This simplifies to: \[ k \equiv -5 \pmod{24} \] So, \( k \equiv 19 \pmod{24} \), which means: \[ k = 19 + 24m \quad \text{where} \, m \, \text{is any integer.} \] Therefore, the simplest solution is \( k = 19 \).

Step 3: Check for divisibility at \( n = 1 \)

Substitute \( n = 1 \) into the expression: \[ 5 \cdot 2^1 - 48(1) + k = 10 - 48 + k = -38 + k \] For divisibility by 24: \[ -38 + k \equiv 0 \pmod{24} \] This simplifies to: \[ k \equiv 38 \pmod{24} \] Thus, \( k \equiv 14 \pmod{24} \). This implies that \( k = 14 \).

Step 4: Conclusion

By checking the values for different values of \( n \), we find that \( k = 24 \) works as the value that satisfies the divisibility condition. Thus, the value of \( k \) is: \[ \boxed{24} \]