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 stillbe true for some integers x with x ‡ −1, 0, 1.

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Answered: Fermat's Little Theorem says that for…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Prove the following statements (using either direct or indirect proof method) state method used (a) For every integer n, n3 + n is even. (Hint: An integer can be either odd or even. Use both cases) (b) The product of two integers, one of the form 3m1 + 1 and the other of the form 3m2 + 2, where m1 and m2 are integers, is of the form 3m3 + 2 for some integer m3.
. (a) If g is a primitive root modulo 37, which of the numbers g², g³,..., g8 are primitive roots modulo 37?
1.) Mathematical induction is useful in determining whether the statement REMAINS TRUE with any natural number as k. Is the statement TRUE or FALSE? 2.) We have only one (1) step that we may utilize in Mathematical induction – the Inductive step. Is this statement TRUE or FALSE? a. Underline only the PART that makes the following statement erroneous: "The Basis step helps in determining whether the first iteration of the statement, which is P(k+10), is TRUE." 3.) In a statement in Mathematical induction, we may only use positive integers or natural numbers for k. Is this statement TRUE or FALSE? 4.) We use k and k+1 for the Inductive step and Basis step respectively. Is this statement TRUE or FALSE? 5.) Prove that given any integer for n, n³ + 2n will be divisible by 3. n²(n+1)? 6.) 13 + 23 + 33 + +n³ ... 4 1 7.) 1(2) 1 + + 2(3) 1 + 3(3) 1 + п(п+1) given that any integer for n is positive. ... n+1' 8.) Write the formal proof that 2"+2 + 32n+1 is divisible by 7 for all positive…
Let p and q be prime numbers.(1) If p is not equal to q, how many generators does Zpq have?(2) If p = q, how many generators does Zp2 have
a. #5 Prove that for all integers n, it is the case that n is even if an only if 3n is even. That is, prove both implications: if n is even, then 3n is even, and if 3n is even, then n is even. b. #7 Consider the statement: for all integers a and b, if a is even and b is a multiple of 3, then ab is a multiple of 6.Then state the converse, tell if it is true, and prove or disprove.
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Suppose that P(n), Q(n) and R(n) are statements about the integer n. Let S(n) be the statement: “If P(n) is true then Q(n) and R(n) are both true." (a) The contrapositive of S(n) is the statement Olf Q (n) and R(n) are both true then P(n) is false. Olf Q (n) and R(n) are both true then P(n) is true. Olf at least one of Q(n) and R(n) is false then P(n) is false. Olf Q (n) and R(n) are both false then P(n) is false. Olf at least one of Q(n) and R(n) is false then P(n) is true. Olf P(n) is false then at least one of Q(n) and R(n) is false. Olf Q (n) and R(n) are both false then P(n) is true. ONone of the other answers. (b) The statement: "If Q (n) and R(n) are both true then P(n) is true." is OThe converse of S(n). OThe contrapositive of S(n). OThe converse of the contrapositive of S(n). ONone of the other answers.
1)For each of the following, find an integer value x which satisfies the equation, or explain why there is no solution to the equation. (a) 2x ≡ 3(mod 25) (b) 6x ≡ 4(mod 15)
Theorem 12.2. For all odd prime numbers p, ((²¹)!)² = (−1)²+¹ mod p. Remark. Apply Theorem 12.1 to the expression ((²¹)!)² to see that the expression is congruent to (p-1)!(-1). Now apply Wilson's theorem to get the result.
Prove that the statement is true for every positive integer n.
Put the steps of a proof for the following claim in the proper order: 4 + 8 + 12 + + 4n = 2n(n + 1) + 4k + 4(k + 1) = 2k (k + 1) +4(k + 1) 2 (k + 1) (k +2) • 4+8+ 12 + .. + 4k = 2k (k + 1). Assume 4 + 8 + 12 + v Let n = 1. Clearly, 4 = 2(1)(1+ 1). v WTS: 4 +8+ 12 + + 4k + 4(k + 1) = 2 (k + 1) ((k + 1) + 1)
Fermat's Little Theorem says that for all natural numbers p and integers x, if p is prime then xº = 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 xP = x mod p can still be true for some integers x with x -1, 0, 1.

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