Step-by-step explanation:
To find the remainder when 3^100 is divided by 8, we can use the concept of modular arithmetic.
First, let's find the remainder when 3^100 is divided by 8. We can do this by calculating the remainder at each step of the exponentiation process.
When we divide 3 by 8, the remainder is 3. Therefore, we can write 3 as 8a + 3, where a is an integer.
Now, let's calculate the powers of 3 modulo 8:
3^2 = (8a + 3)^2 = 64a^2 + 48a + 9 ≡ 1a + 1 (mod 8)
3^3 = 3 * 3^2 ≡ 3 * (1a + 1) ≡ 3a + 3 (mod 8)
3^4 = (3^2)^2 ≡ (1a + 1)^2 ≡ 1a^2 + 2a + 1 ≡ 2a + 1 (mod 8)
3^5 = 3 * 3^4 ≡ 3 * (2a + 1) ≡ 6a + 3 (mod 8)
3^6 = (3^2)^3 ≡ (1a + 1)^3 ≡ 1a^3 + 3a^2 + 3a + 1 ≡ 3a + 1 (mod 8)
3^7 = 3 * 3^6 ≡ 3 * (3a + 1) ≡ 7a + 3 (mod 8)
3^8 = (3^2)^4 ≡ (1a + 1)^4 ≡ 1a^4 + 4a^3 + 6a^2 + 4a + 1 ≡ 4a + 1 (mod 8)
As we can observe, the powers of 3 repeat in a pattern: 1, 3, 1, 3, 1, 3, 1, 3, ...
Since 100 is a multiple of 4 (100 = 4 * 25), we can conclude that 3^100 ≡ 1 (mod 8).
Therefore, the remainder when 3^100 is divided by 8 is 1.