Can you do #5? ### 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) \( 99^{999} \mod 31 \)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 \( (a + b)^p \equiv a + b \pmod{p} \).**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 \( 2^r \cdot k + 1 \).**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 \( n \).11. **Show that any proper factor, whether prime or not, of a composite Mersenne number \( 2^p - 1 \) is of the form \( 1 + 2pk \) for some \(

Name: Can you do #5? ### Exercises for Section 4.11. **Use Fermat's Theorem
Uploaded: 2024-03-20T07:06:41.000Z
Channel: Bartleby
Description: Can you do #5? ### 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) \( 99^{999} \mod 31 \)2. Show that \( 11^{84} - 5^{84} \) is divisible