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 for the least common multiple of a and b is lcm(a,b). For example, lcm(24,30)=120. Please do the following: (a) Develop a formula for the least common multiple of two positive integers in terms of their prime factorizations; your formula should be similar to the in Theorem 36.5 (b) Use your formula to show: If a and b are positibe integers, then ab=gcd(a,b)lcm... click for more