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

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

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