Question

When $10^{100}$ is divided by 7, the remainder is ?

• A. 6
• B. 3
• C. 1
• D. 4

Answer 1

The Correct Option is D

To find the remainder when \(10^{100}\) is divided by 7, we will use Fermat's Little Theorem. Fermat's Little Theorem 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, \(a = 10\) and \(p = 7\).

Step 1: Apply Fermat's Little Theorem

\[ 10^6 \equiv 1 \pmod{7} \]

Step 2: Break down the exponent

\[ 100 = 6 \times 16 + 4 \] \[ 10^{100} = (10^6)^{16} \times 10^4 \]

Step 3: Simplify using the theorem

\[ (10^6)^{16} \equiv 1^{16} \equiv 1 \pmod{7} \] \[ 10^{100} \equiv 10^4 \pmod{7} \]

Step 4: Compute \(10^4 \mod 7\)

\[ 10^2 = 100 \quad \Rightarrow \quad 100 \div 7 = 14 \, \text{remainder } 2 \] \[ 10^2 \equiv 2 \pmod{7} \]

Now:

\[ 10^4 = (10^2)^2 \equiv 2^2 \equiv 4 \pmod{7} \]

Final Answer: The remainder when \(10^{100}\) is divided by 7 is 4.

Answer 2

We are asked to find the remainder when \(10^{100}\) is divided by 7. This is equivalent to finding:

\[ 10^{100} \pmod{7} \]

Step 1: Apply Fermat's Little Theorem

Since 7 is prime, by Fermat's Little Theorem:

\[ 10^6 \equiv 1 \pmod{7} \]

Step 2: Reduce the exponent

We divide 100 by 6:

\[ 100 \div 6 = 16 \, \text{ remainder } 4 \]

Thus:

\[ 10^{100} \equiv 10^4 \pmod{7} \]

Step 3: Compute \(10^4 \mod 7\)

\[ 10^4 = 10000 \] \[ 10000 \div 7 = 1428 \, \text{ remainder } 4 \]

Final Answer:

The remainder when \(10^{100}\) is divided by 7 is 4.