Step 1: Understanding the Concept:
We are given that an expression involving \( n \) is always divisible by 24. We can use the property of congruences or simply substitute values of \( n \) (like \( n=1 \)) to find the required value of \( k \).
Step 2: Key Formula or Approach:
Method 1: Substitution (easiest for multiple choice).
Method 2: Modular Arithmetic.
\( 5^{2n} = 25^n \equiv 1^n \equiv 1 \pmod{24} \).
Step 3: Detailed Explanation:
Method 1: Using \( n=1 \)
Since the statement holds for all \( n \in \mathbb{N} \), it must hold for \( n=1 \).
\[ E = 5^{2(1)} - 48(1) + k \]
\[ E = 25 - 48 + k \]
\[ E = k - 23 \]
For \( E \) to be divisible by 24, \( k - 23 \) must be a multiple of 24.
\[ k - 23 = 24m \]
For the least positive integral value, let \( m=0 \):
\[ k - 23 = 0 \implies k = 23 \]
Let's verify for \( n=2 \) with \( k=23 \):
\[ E = 5^4 - 48(2) + 23 = 625 - 96 + 23 \]
\[ E = 529 + 23 = 552 \]
Is 552 divisible by 24?
\[ 552 \div 24 = 23 \]
Yes, it is divisible.
Method 2: Modular Arithmetic
Expression \( f(n) = 25^n - 48n + k \).
Modulo 24:
\[ 25 \equiv 1 \pmod{24} \implies 25^n \equiv 1 \pmod{24} \]
\[ 48n \equiv 0 \pmod{24} \]
So, \( f(n) \equiv 1 - 0 + k \pmod{24} \)
\[ f(n) \equiv 1 + k \pmod{24} \]
For \( f(n) \) to be divisible by 24, the remainder must be 0.
\[ 1 + k \equiv 0 \pmod{24} \]
\[ k \equiv -1 \equiv 23 \pmod{24} \]
The least positive integer is 23.
Step 4: Final Answer:
The least positive integral value of \( k \) is 23.