In parts (a)- (e), involve the theorems of Fermat, Euler, Wilson, and the Euler Phi-function.(a) Show (4(29!) + 5!) = 0 mod 31(b) Prove a21 = a mod 15 for all integers a(c) If p,q are distinct primes and ged(a, p) = ged(a, q) = 1, prove a-1)(4-1) = 1 mod pq%3D%3D(d) Prove 384k+5 = -2 mod 49 for all integers k(e) Suppose a, b, c, d are integers. Show 8 (2 3 5c)

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Answered: In parts (a)- (e), involve the theorems…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

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When we studied prime numbers we looked at the Euler phi-function ∅(n) = the number of integers between 1 and n with no factors in common with n = n – (the number of integers between 1 and n with a factor in common with n)Look at the special case where n = pq, with p and q being different primes.Let A = {the integers between 1 and pq which are divisible by p} and argue that |A| = qLet B = {the integers between 1 and pq which are divisible by q} and argue that |B| = pWhy is |A ∩ B| = 1?Conclude that |A U B| =p + q -1 and that ∅(pq) = pq - (p + q -1) = (p-1)(q-1)
Only Question 4
Fermat's Little Theorem says that for all natural numbers p and integers x, if p is prime then x² = x mod p. (i) Show that the statement is wrong, if we drop the assumption that p is prime. [Remember: All you need to do is to give a counterexample.] (ii) Show, by giving an example, that if p is not prime, then x² = x modp can still be true for some integers x with x ‡ −1, 0, 1.
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(10) Prove that for any prime number P>3. p² = 1 (mod 24) (Hint: chinese remainder theorem tells us SP² = 1 (mod 3) p² = 1 (mod 8) need to show we only need to p²=1 (mod 8), which we know that P PH two integers. The hard part is to show means S1 p²2-1 81 = (P-1) (PH). must be odd. Hence, P-1, So it is enough to show 2 | P-1 F are PH).

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In parts (a)- (e), involve the theorems of Fermat, Euler, Wilson, and the Euler Phi-function. (a) Show (4(29!) +5!) = 0 mod 31 (b) Prove a21 = a mod 15 for all integers a (c) If p, q are distinct primes and ged(a, p) = ged(a, q) = 1, prove a-1)(4-1) = 1 mod pq %3D 110 for all integers k