Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Question

Using Wilson's Theorm, Fermat's Little Theorm and The Pollard Factorization Metod how could I solve Quetion 2 in Section 6.1 

for by using other than 2 as the In the Pollard reader can verify), but 1, 5, = 5, = 2269. It follows Example 6.7. rk the least positive residue of 2k me Example 6.6. We compute (r - 1, 5,157,437) at each step. requires nine steps, because (r-1, 5,157,437) = 1 for k =1, 2, 3, 4, 5, 67437 reader can verify), but (ro-1, 5, 157,437) = (4,381,439, 5, 157,437) = 2269 (as the that 2269 is a divisor of 5,157,437. method on of 2 as the base, we can the and find a factor The Pollard does not nothing in the method is after by small but the of such %3D method depends on the choice of 2 as the base, we can extend the method and find in the p-1 a for more integers by using integers other than 2 as the base. In practice, the Pollardclor methods as the quadratic sieve and the elliptic curve method. 6.1 EXERCISES 1. Show that 10!+1 is divisible by 11, by grouping together pairs of inverses modulo 114 occur in 10! 2. Show that 12!+1 is divisible by 13, by grouping together pairs of inverses modulo 13 thes occur in 12!. 3. What is the remainder when 16! is divided by 19? 4. What is the remainder when 5!25! is divided by 31? 5. Using Wilson's theorem, find the least positive residue of 8 · 9 · 10 - 11-12- 13 modulo 7 6. What is the remainder when 7·8 ·9. 15· 16 17· 23 · 24 · 25 43 is divided by 11? 7. What is the remainder when 18! is divided by 437? 8. What is the remainder when 40! is divided by 1763? 9. What is the remainder when 5100 is divided by 7? 10. What is the remainder when 62000 is divided by 11? 11. Using Fermat's little theorem, find the least positive residue of 3999,999,999 modulo 7. 12. Using Fermat's little theorem, find the least positive residue of 21,000,000 modulo 17. 13. Show that 310 = 1 (mod 11²). 14. Using Fermat's little theorem, find the last digit of the base 7 expansion of 3. 15. Using Fermat's little theorem, find the solutions of the following linear congruences. a) 7x = 12 (mod 17) b) 4x = 11 (mod 19) 16. Show that if n is a composite integer with n 4, then (n- 1)!=0 (mod n). 17. Show that if p is an odd prime, then 2(p - 3)! = -1 (mod p).
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Transcribed Image Text:for by using other than 2 as the In the Pollard reader can verify), but 1, 5, = 5, = 2269. It follows Example 6.7. rk the least positive residue of 2k me Example 6.6. We compute (r - 1, 5,157,437) at each step. requires nine steps, because (r-1, 5,157,437) = 1 for k =1, 2, 3, 4, 5, 67437 reader can verify), but (ro-1, 5, 157,437) = (4,381,439, 5, 157,437) = 2269 (as the that 2269 is a divisor of 5,157,437. method on of 2 as the base, we can the and find a factor The Pollard does not nothing in the method is after by small but the of such %3D method depends on the choice of 2 as the base, we can extend the method and find in the p-1 a for more integers by using integers other than 2 as the base. In practice, the Pollardclor methods as the quadratic sieve and the elliptic curve method. 6.1 EXERCISES 1. Show that 10!+1 is divisible by 11, by grouping together pairs of inverses modulo 114 occur in 10! 2. Show that 12!+1 is divisible by 13, by grouping together pairs of inverses modulo 13 thes occur in 12!. 3. What is the remainder when 16! is divided by 19? 4. What is the remainder when 5!25! is divided by 31? 5. Using Wilson's theorem, find the least positive residue of 8 · 9 · 10 - 11-12- 13 modulo 7 6. What is the remainder when 7·8 ·9. 15· 16 17· 23 · 24 · 25 43 is divided by 11? 7. What is the remainder when 18! is divided by 437? 8. What is the remainder when 40! is divided by 1763? 9. What is the remainder when 5100 is divided by 7? 10. What is the remainder when 62000 is divided by 11? 11. Using Fermat's little theorem, find the least positive residue of 3999,999,999 modulo 7. 12. Using Fermat's little theorem, find the least positive residue of 21,000,000 modulo 17. 13. Show that 310 = 1 (mod 11²). 14. Using Fermat's little theorem, find the last digit of the base 7 expansion of 3. 15. Using Fermat's little theorem, find the solutions of the following linear congruences. a) 7x = 12 (mod 17) b) 4x = 11 (mod 19) 16. Show that if n is a composite integer with n 4, then (n- 1)!=0 (mod n). 17. Show that if p is an odd prime, then 2(p - 3)! = -1 (mod p).
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