If 17²⁰⁰ divided by 19 the remainder is:

To find the remainder when \( 17^{200} \) is divided by 19, we can use **Fermat's Little Theorem**. 

Fermat's Little Theorem states that if \( p \) is a prime number and \( a \) is any integer that is not divisible by \( p \), then:

\[
a^{p-1} \equiv 1 \ (\text{mod}\ p)
\]

In this case:
- \( a = 17 \)
- \( p = 19 \)

Thus, by Fermat's Little Theorem:

\[
17^{18} \equiv 1 \ (\text{mod}\ 19)
\]

Now, we need to express \( 17^{200} \) in terms of powers of 17 modulo 19. To do this, divide 200 by 18 and find the remainder:

\[
200 \div 18 = 11 \text{ remainder } 2
\]

So, \( 17^{200} \equiv 17^2 \ (\text{mod}\ 19) \).

Now, calculate \( 17^2 \ (\text{mod}\ 19) \):

\[
17^2 = 289
\]
\[
289 \div 19 = 15 \text{ remainder } 4
\]

Thus, the remainder when \( 17^{200} \) is divided by 19 is **4**.

Would you like more details or have any questions?

Here are 5 related questions to explore further:
1. How does Fermat's Little Theorem apply to other primes?
2. What is the significance of modular arithmetic in number theory?
3. Can you prove Fermat’s Little Theorem for any specific prime?
4. How do you calculate large powers efficiently using modular arithmetic?
5. What other methods exist to find remainders besides Fermat's Little Theorem?

**Tip**: When dealing with large powers in modular arithmetic, breaking them into smaller cycles (as in Fermat's theorem) helps simplify calculations.

Discover how to calculate the remainder when 17²⁰⁰ is divided by 19 using Fermat's Little Theorem and modular arithmetic. Follow a detailed step-by-step breakdown suitable for students in Grades 10-12.

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