Answers to practice problems with prime numbers.

1. What is the most important thing (the most useful property) about prime numbers?

Every natural number greater than 1 can be factored into a product of prime numbers in one and only one way.

2. Find the prime factorization of each of these numbers:

a. 308 = 2*2*7*11

b. 975 = 3*4*4*13

c. 343 = 7*7*7

d. 170 = 2*5*17

3. Write down the first 10 prime numbers

2, 3, 5, 7, 11, 13, 17, 19, 23, 29

4. Determine whether each of these numbers is prime. If it isn't prime, give its prime factorization. If it is prime, tell what numbers you tried dividing by to determine that it is prime.

a. 181 Yes, it's prime. I checked 2, 3, 5, 7, 11, 13, 17

b. 231 = 3*7*11

c. 323 = 18*19

d. 401 Yes, it's prime. I checked 2, 3, 5, 7, 11, 13, 17 and 23

5. Find the remainder for each of these division problems:

a. 13,691 ÷ 167 The remainder is 164

b. 287,911 ÷ 211 The remainder is 107

c. 59,621 ÷ 97 The remainder is 63

d. 78,443 ÷ 577 The remainder is 548