Final answer:
The remainder of 3¹⁰⁰ when divided by 7 is found using modular arithmetic, yielding a remainder of 4.
Step-by-step explanation:
To find the remainder of 3¹⁰⁰ when divided by 7, we can use modular arithmetic and properties of exponents. The idea is to find a pattern in the powers of 3 modulo 7. Observe that:
- 3¹ mod 7 = 3
- 3² mod 7 = 2
- 3³ mod 7 = 6
- 3⁴ mod 7 = 4
- 3⁵ mod 7 = 5
- 3⁶ mod 7 = 1
- 3⁷ mod 7 = 3 (since 3⁶ ⋅ 3 mod 7 = 1⋅ 3 = 3)
Notice that every power of 3 raised to a multiple of 6 gives a remainder of 1 when divided by 7. Consequently, 3⁶⁹ mod 7 = 1, as 69 is a multiple of 6. We can therefore write 3¹⁰⁰ as (3⁶)¹⁶⋅ 3⁴, and applying the rule:
(3⁶¹⁶ mod 7) ⋅ (3⁴ mod 7) = (1¹⁶ mod 7) ⋅ (4 mod 7)
= 1 ⋅ 4
= 4
Therefore, the remainder of 3¹⁰⁰ when divided by 7 is 4.