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 \(p = 7\).

According to Fermat's Little Theorem:

\(10^{6} \equiv 1 \pmod{7}\) 

We need to find \(10^{100} \mod 7\). First, express \(100\) in terms of the exponent 6:

\(100 = 6 \times 16 + 4\)

Therefore:

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

Because \(10^6 \equiv 1 \pmod{7}\), it follows that:

\((10^6)^{16} \equiv 1^{16} \equiv 1 \pmod{7}\)

So:

\(10^{100} \equiv 10^4 \pmod{7}\)

We need to calculate \(10^4 \mod 7\):

Calculate \(10^2 \equiv 2 \pmod{7}\) because:

\(10^2 = 100\) and \(100 \div 7 = 14\) remainder \(2\).

Now, calculate \(10^4 = (10^2)^2 \equiv 2^2 \pmod{7}\).

\(2^2 = 4\), therefore:

\(10^4 \equiv 4 \pmod{7}\)

Thus, 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 $10^{100} \pmod{7}$.
By Fermat's Little Theorem, since 7 is prime:

$10^6 \equiv 1 \pmod{7}$

So, we can reduce $10^{100} \pmod{7}$ by dividing 100 by 6 (since the powers of 10 repeat every 6 terms modulo 7):

$100 \div 6 = 16 \text{ remainder } 4$

Thus:

$10^{100} \equiv 10^4 \pmod{7}$

Now calculate $10^4 \pmod{7}$:

$10^4 = 10000 \implies 10000 \div 7 = 1428 \text{ remainder } 4$

Thus, the remainder when $10^{100}$ is divided by 7 is 4.