Let n be a positive integer. Prove that if n is composite, then n has a...

Let n be a positive integer. Prove that if n is composite, then n has a prime factor less than or equal to sqrt(n) . (Hint: first show that n has a factor less than or equal to sqrt(n) )

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Colby Messinger answered 9 months ago

(a) Let a > 1 be an integer. Prove that any composite divisor of a −...

(a) Let a > 1 be an integer. Prove that any composite divisor of a − 1 is a pseudoprime of base a. (b) Suppose, for some m, than n divides a^(m − 1) and n ≡ 1 (mod m). Prove that if n is composite, then n is a pseudoprime of base a. (c) Use (b) to give two examples pseudoprimes of base a with a = 2 and a = 3 (hint: take m = 2k to be...

Let G be an abelian group and n a fixed positive integer. Prove that the following...

Let G be an abelian group and n a fixed positive integer. Prove that the following sets are subgroups of G. (a) P(G, n) = {gn | g ∈ G}. (b) T(G, n) = {g ∈ G | gn = 1}. (c) Compute P(G, 2) and T(G, 2) if G = C8 × C2. (d) Prove that T(G, 2) is not a subgroup of G = Dn for n ≥ 3 (i.e the statement above is false when G is...

Prove the following theorem. If n is a positive integer such that n ≡ 2 (mod...

Prove the following theorem. If n is a positive integer such that n ≡ 2 (mod 4) or n ≡ 3 (mod 4), then n is not a perfect square.

Prove or disprove that 3|(n 3 − n) for every positive integer n.

Prove that τ(n) < 2 n for any positive integer n. This is a question in...

Prove that τ(n) < 2 n for any positive integer n. This is a question in Number theory

Let A[1..n] be an array of distinct positive integers, and let t be a positive integer....

Let A[1..n] be an array of distinct positive integers, and let t be a positive integer. (a) Assuming that A is sorted, show that in O(n) time it can be decided if A contains two distinct elements x and y such that x + y = t. (b) Use part (a) to show that the following problem, re- ferred to as the 3-Sum problem, can be solved in O(n2) time: 3-Sum Given an array A[1..n] of distinct positive integers, and...

8.Let a and b be integers and d a positive integer. (a) Prove that if d...

8.Let a and b be integers and d a positive integer. (a) Prove that if d divides a and d divides b, then d divides both a + b and a − b. (b) Is the converse of the above true? If so, prove it. If not, give a specific example of a, b, d showing that the converse is false. 9. Let a, b, c, m, n be integers. Prove that if a divides each of b and c,...

Let t be a positive integer. Prove that, if there exists a Steiner triple system of...

Let t be a positive integer. Prove that, if there exists a Steiner triple system of index 1 having v varieties, then there exists a Steiner triple system having v^t varieties

Use induction to prove that for any positive integer n, 8^n - 3^n is a multiple...

Use induction to prove that for any positive integer n, 8^n - 3^n is a multiple of 5.

Let n be a positive integer. Let S(n) = n sigma j=1 ((1/3j − 2) −...

Let n be a positive integer. Let S(n) = n sigma j=1 ((1/3j − 2) − (1/3j + 1)). a) Compute the value of S(1), S(2), S(3), and S(4). b) Make a conjecture that gives a closed form (i.e., not a summation) formula for the value of S(n). c) Use induction to prove your conjecture is correct.

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