What is the greatest common divisor of 4655 and 17689? Prove that the remainder of the division by 9 of a positive integer written in the decimal system is equal to the remainder of division by 9 of the sum of its digits.

Discrete Math Test 3

Yoram Sagher
4/17/2023

Your solutions should be submitted through Canvas in .pdf in 60 minutes.

1. What is the greatest common divisor of 4655 and 17689?. An answer without a relevant calculation will receive no credit.
2. Find all x so that |x + 1| + 3 |x − 2| < 5. An answer without a relevant calculation will receive no credit.
3. Prove that the remainder of the division by 9 of a positive integer written in the decimal system is equal to the remainder of division by 9 of the sum of its digits.
4. Find irrational numbers, a, b so that ab is a rational number.
5. A prime number is an integer whose only divisors are 1 and itself. Prove that if p is a prime and n is any integer that is not a multiple p then the greatest common divisor of n and p is 1.
6. We proved that if s and t are positive integers and the greatest common divisor of s and t is d, then there exist integers, x and y so that xs+yt = d. Prove that if p.is a prime number and s and t are positive integers so that and p divides s·t but does not divide s, then p divides t.