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.

Discrete Math

Discrete Math Final Spring 2023

   Yoram Sagher

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

1. Find all x so that |x 1| 3|x 2| 7|x 4| 16
2. 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
3. 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
4. Prove that 447 is an irrational number
5. 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
6. Find integers x, y do that x 1001 y 385 gcd1001, 385.
7. 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.
8. 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 .
9. Prove that if p 2 is a prime number then 2p−1 − 1 is a multiple of p.
10. Prove that if p is a prime number and n a positive integer then n p − n is divisible by p.
11. Prove that
    ∑
    k1
    n
    k n
    k n2n−1
12. Prove that
    ∑
    k1
    n
    −1k k n
    k 0.
13. Prove that
    n − 1
    k − 1 n − 1
    k n
    k
14. Prove that

n − 2

k − 2 2 n − 2
k − 1 n − 2
k n
k

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