Question

The remainder when $6^{1029}$ is divided by 7 is:

• A. 6
• B. 1
• C. 0
• D. 3

Answer 1

Answer: (A)

6

Answer 2

We are tasked to compute $6^{1029} \mod 7$. Using Fermat's Little Theorem:

Step 1: Apply Fermat's Little Theorem.

Fermat's Little Theorem states:

$a^{p-1} \equiv 1 \pmod{p},$

for a prime p and an integer a not divisible by p. Here $a = 6$ and $p = 7$. Since 6 is not divisible by 7, we have:

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

Step 2: Simplify the exponent.

To compute $6^{1029} \mod 7$, divide 1029 by 6 (the exponent cycle length from Fermat's theorem):

$1029 \div 6 = 171 \text{ remainder 3}.$

Thus:

$6^{1029} \equiv 6^3 \pmod{7}.$

Step 3: Compute $6^3 \mod 7$.

Now calculate $6^3 \mod 7$:

$6^3 = 216.$

Find the remainder when 216 is divided by 7:

$216 \div 7 = 30 \text{ remainder 6}.$

Thus:

$6^3 \equiv 6 \pmod{7}.$

Final Answer:

$6$