Question

Value of 2⁴⁸ (mod 15) is :

• A. 1
• B. 0
• C. -1
• D. 7

Answer 1

The Correct Option is A

To solve for \(2^{48} \mod 15\), we use properties of modular arithmetic and the Euler's Theorem. Euler's Theorem states that if \(a\) and \(n\) are coprime, then \(a^{\phi(n)} \equiv 1 \pmod{n}\), where \(\phi(n)\) is the Euler's totient function.

1. Calculate \(\phi(15)\):

\(\phi(15) = \phi(3 \times 5) = \phi(3) \times \phi(5) = (3-1) \times (5-1) = 2 \times 4 = 8\)

2. Since 2 and 15 are coprime, by Euler's Theorem:

\(2^8 \equiv 1 \pmod{15}\)

3. Now express 48 in terms of multiples of 8:

\(48 = 8 \times 6\)

4. Apply the exponentiation rule:

\(2^{48} = (2^8)^6\)

5. Substitute the congruence:

\((2^8)^6 \equiv 1^6 \equiv 1 \pmod{15}\)

Therefore, \(2^{48} \equiv 1 \pmod{15}\).

Hence, the value of \(2^{48} \mod 15\) is 1.