**Math**

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.

# 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.

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