Prove that if an integer n is not divisible by 3, then n^2=3k+1 for some integer k.
Note: “not divisible by 3” means either “n=3m+1 for some integer m” or “n=3m+2 for some integer m”.
Prove that if an integer n is not divisible by 3, then n^2=3k+1 for some integer...
1. Let n be a positive integer with n > 1000. Prove that n is divisible by 8 if and only if the integer formed by the last three digits of n is divisible by 8.
1) Let n and m be positive integers. Prove: If nm is not divisible by an integer k, then neither n norm is divisible by k. Prove by proving the contrapositive of the statement. Contrapositive of the statement:_ Proof: Direct proof of the contrapositive
Prove that a(a + 1)(2a + 1) is divisible by 6 for integer a using a quicker proof of this, based on the observation that 6 I m (6 divides m) if and only if 2 I m (2 divides m) and 3 I m (3 divides m). Please use modulo congruences.
Prove by Contraposition: Vb e Z, if 362 – 2b is not divisible by some integer a, then b is not divisible by a. For full credit, you must write the contrapositive before you start the proof.
4 Mathematical Induction 1. Prove that 1.1!+2-2!+3-3! +...+n.n! = (n+1)!- 1 for every integer n> 1. 2. Prove that in > 0, n - n is divisible by 5. 3. Prove that 'n > 0,1-21 +222 +3.23 + ... + n.2n = (n-1). 2n+1 +2.
33. Prove that 11n - 6 is divisible by 5 for every positive integer n.
please answer all the questions.
just rearranging. Explanation is not needed.
Use modular arithmetic to prove that 3|(221 – 1) for an integer n > 0. Hence, 3|(221 – 1) for n > 0. To show that 3|(221 – 1), we can show that (221 – 1) = 0 (mod 3). We have: (221 – 1) = (4” – 1) (mod 3) Then, (22n – 1) = (1 - 1) = 0 (mod 3) Since 4 = 1 (mod 3),...
Tems.] Use the second principle of induction to prove that every positive integer n has a factorization of the form 2m, where m is odd. (Hint: For n > 1, n is either odd or is divisible by 2.)
prove the product of 4 consecutive integers is always divisible by 24 using the principles of math induction. Could anyone help me on this one? Thanks in advance!Sure For induction we want to prove some statement P for all the integers. We need: P(1) to be true (or some base case) If P(k) => P(k+1) If the statement's truth for some integer k implies the truth for the next integer, then P is true for all the integers. Look at...
5. (a) Let m,n be coprime integers, and suppose a is an integer which is divisible by both m and n. Prove that mn divides a. (b) Show that the conclusion of part (a) is false if m and n are not coprime (ie, show that if m and n are not coprime, there exists an integer a such that mla and nla, but mn does not divide a). (c) Show that if hef(x,m) = 1 and hcf(y,m) = 1,...