Proof by Induction : Prove that 3^(3n-2) + 3^(6n+1) +1 can be divided by 13 with no remainder for any integer n.

Prove that 3^(3n-2) + 3^(6n+1) +1 can be divided by 13 with no remainder for any integer n.

Greatest common divisor

Find the greatest common divisor of a) 705 and 1962 b) 5339 and 2565

Solving equations using Euclidean algorithm

(1) 234x+539y=1 (2) 875x+235y=10 --- (See attached file for full problem description)

Least common multiple

Find the least common multiple of 2345 and 5236, 10000 and 10001.

Induction proof

Let n ≥ 2 and k be any positive integers. Prove that (n – 1) | (nk – 1). (We can use induction.) (Please show each step of your solution. Thank you.) --- (See attached file for full problem description)

Greatest common divisor proof

Prove that (a,b,c)=((a,b)c) --- (See attached file for full problem description)

Greatest common divisor

Suppose that n is an integer >1 and a,b are positive integers. Prove... --- (See attached file for full problem description)

Logic problem and divisors

Suppose you have 2 pitchers... --- (See attached file for full problem description)

Find Prime Factorization (by hand) of 2006 and 2007.

Find Prime factorization (by hand) of 2006 and 2007.

Prime Factorization : Proof - Square-Free Integers

Show that every positive integer n can be written in the form n = ab where a is square-free and b is a square. Show that b is thi the largest square dividing n. (A square-free integer is an integer that is not divisible by any square > 1).