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 Pollardreader can verify), but 1, 5, = 5, = 2269. It followsExample 6.7.rk the least positive residue of 2k meExample 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, 67437reader can verify), but (ro-1, 5, 157,437) = (4,381,439, 5, 157,437) = 2269 (as thethat 2269 is a divisor of 5,157,437.method on of 2 as the base, we can the and find a factorThe Pollard does not nothing in themethod is after by small but the of such%3Dmethod depends on the choice of 2 as the base, we can extend the method and find in thep-1afor more integers by using integers other than 2 as the base. In practice, the Pollardclormethods as the quadratic sieve and the elliptic curve method.6.1 EXERCISES1. Show that 10!+1 is divisible by 11, by grouping together pairs of inverses modulo 114occur in 10!2. Show that 12!+1 is divisible by 13, by grouping together pairs of inverses modulo 13 thesoccur 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 76. 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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To show that 12!+1 is divisible by 13, by grouping together pairs of inverse modulo 13 that occur in…

occur in 10! 2. Show that 12!+1 is divisible by 13, by grouping together pairs of inverses modulo 13 that CHr in 12!