Can you do #9? ### Exercises for Section 4.11. **Use Fermat's Theorem to compute the following quantities:** - (a) \( 31^{100} \mod 19 \) - (b) \( 2^{10000} \mod 29 \) - (c) \( 9^{999} \mod 31 \)2. **Show that** \( 11^{84} - 5^{84} \) **is divisible by 7.**3. **Show that if** \( n \equiv 2 \pmod{4} \), **then** \( 9^n + 8^n \) **is divisible by 5.**4. **For which values of** \( n \) **is** \( 3^n + 2^n \) **divisible by 13? By 7?**5. **Use Fermat's Theorem to show that** \( n^{13} - n \) **is divisible by 2730 for all** \( n \).6. **Show that if** \( p > 3 \) **is prime, then** \( ab^p - ba^p \) **is divisible by** \( 6p \).7. **Show, using the Binomial Theorem, that if** \( p \) **is prime and** \( a \) **and** \( b \) **are integers, then** \( (a+b)^p \equiv a^p + b^p \pmod{p} \).8. **Show that no prime number of the form** \( 4k + 3 \) **can divide a number of the form** \( n^2 + 1 \).9. **Show that there are infinitely many primes of the form** \( 16k + 1 \). **More generally, show that for any** \( r > 0 \), **there are infinitely many primes of the form** \( 2^rk + 1 \).10. **Let** \( n = r^4 + 1 \). **Show that** \( 3, 5, \) **and** \( 7 \) **cannot divide** \( n \). **What is the smallest prime that can divide** \( n \)? **Determine the form of the prime divisors of** \(

Name: Can you do #9? ### Exercises for Section 4.11. **Use Fermat's Theorem
Uploaded: 2024-03-20T07:06:41.000Z
Channel: Bartleby
Description: Can you do #9? ### Exercises for Section 4.11. **Use Fermat's Theorem to compute the following quantities:** - (a) \( 31^{100} \mod 19 \) - (b) \( 2^{10000} \mod 29 \) - (c) \( 9^{999} \mod 31 \)2. **Show that** \( 11^{84} - 5^{84} \) **is divi