Discrete Math Final Spring 2023

Your solutions should be submitted through Canvas in .pdf in 150 minutes.

Find all x so that |x 1| 3|x 2| 7|x 4| 16

Prove that if r1 an r2 are rational numbers and r1 r2 then there are irrational numbers , x1 and x2 so that r1 x1 x2 r2

Let a, b and q be positive integers. Prove that a and b have the same remainder when divided by q if and only if a − b is a multiple of q

Prove that 447 is an irrational number

Let a and b be positive integers, we denote by gcda, b the largest integer that divides both a and b. Prove that if k ≤ a b is an integer, then gcda, b gcda − kb, b

Find integers x, y do that x 1001 y 385 gcd1001, 385.

Prove that if p is a prime number and x, y, z are positive integers and p divides x y z then it divides at least one of x, y, z.

We denote by a ∨ b the larger of the two numbers, a and b, and by a ∧ b the smaller of the two numbers. Prove that
a ∨ b a b
2 |a − b|
2 and a ∧ b a b
2 − |a − b|
2 .

Prove that if p 2 is a prime number then 2p−1 − 1 is a multiple of p.

Prove that if p is a prime number and n a positive integer then n p − n is divisible by p.

Prove that
∑
k1
n
k n
k n2n−1

Prove that
∑
k1
n
−1k k n
k 0.

Prove that
n − 1
k − 1 n − 1
k n
k

Prove that

n − 2

k − 2 2 n − 2
k − 1 n − 2
k n
k

Let a 0 and b 0. loga b is the number so that aloga b b Prove that  loga blogb a 1