Question

When \(8^2000\) is divided by 7, the remainder equals

• A. 6
• B. 4
• C. 1
• D. None of these

Hint:

When 8^2000 is divided by 7, the remainder equals

Answer 1

The Correct Option is C

To find the remainder when \(8^{2000}\) is divided by 7, we can use Fermat's Little Theorem, which states that if \(p\) is a prime and \(a\) is an integer not divisible by \(p\), then \(a^{p-1} \equiv 1 \pmod{p}\).

Here, \(a = 8\) and \(p = 7\). According to Fermat's Little Theorem:

\[8^{7-1} \equiv 1 \pmod{7}\]

\[8^6 \equiv 1 \pmod{7}\]

Thus, the exponent 2000 can be reduced modulo 6:

\(2000 \mod 6\) equals 2, because:

\(2000 \div 6 = 333\) with a remainder of 2.

Therefore, \(8^{2000} \equiv 8^2 \pmod{7}\).

Calculating \(8^2\):

\(8^2 = 64\).

Finding \(64 \pmod{7}\):

\(64 \div 7 = 9\) with a remainder of \(1\).

Hence, \(8^{2000} \equiv 1 \pmod{7}\).

The remainder when \(8^{2000}\) is divided by 7 is therefore 1.