Question

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

• A. 6
• B. 3
• C. 1
• D. 4

Answer 1

Answer: (D)

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.