Eight people are attending a seminar...

6. Eight people are attending a seminar in a room with eight chairs. In the middle of the seminar, there is a break and everyone leaves the room... (see attachment for rest of question)

Divisors and relative primes

Let a be an integer. Prove that 2a + 1 and a^2+ 1 are relatively prime. ( relative primes are numbers that their largest common divisor is 1).

Discrete Structures - Solving Systems of Equations

Solve the following systems of equations: (a) x=4 (5) and x=7 (11) (b) 3=34 (100) and x=-1 (51) *Please see attachment for proper symbols and complete instructions

Factor Positive Integrers into Primes

Factor into primes the following positive integers: (a) 25 (b) 4200 (c) 10(to the exponent)10 (d) 19 (e) 1 *Please see attachment for proper citation and complete instructions

Lowest Common Multiple (Prime Factorizations)

Let a and b be integers. A common multiple of a and b is an integer n for which a|n and b|n. We call an integer m the least common multiple of n provided (1) m is positive, (2) m is a common multiple of a and b, and (3) if n is any other positive common multiple of a and b, then n [greater than or equal to] m. The notation fo ...continues

Perfect Integers

An integer 'n' is called 'perfect' if it equals the sum of all its divisors 'd' ... {see attachment for complete definition and example} Let 'a' be a positive integer. Prove ... {see attachment}

Perfect Square; Perfect Cube; Perfect Fifth Power

Find the smallest positive integer N such that N/2 is a perfect square, N/3 is a perfect cube and N/5 is a perfect fifth power.

Euclid's Algorithm for Greatest Common Divisor

1. (a) Use the Euclidcan Algorithm to find the greatest common divisor of 13 and 21 (b) and (c) in attachment (PLEASE SEE ATTACHMENT FOR EXPLANATION OF EUCLID'S ALGORITHM AND COMPLETE PROBLEM)

Shuffle and Cardinality

(Bijection, shuffle, cardinality) Let (SYMBOL) = {0,1,2,3...N-1} where N is a positive integer ... (PLEASE SEE ATTACHMENT FOR COMPLETE PROBLEM)

Basic Algebra (subsitution)

Problem: Five children collect N pieces of Halloween candy and decide to split it evenly among them. When they try to divide it they have two pieces of candy left over. One of the children leaves, taking the 26 pieces of candy she collected with her. The remaining four children try to split the N-26 remaining pieces of candy ...continues