1. Determine whether each of the following integers is prime. (Answer yes for prime, no for not prime.)

    1. 19

      Yes

    2. 71

      Yes

    3. 143

      No. (143 = 11×13)

    4. 97

      Yes

    5. 1,113

      No. (1113 = 3×7×53)

  2. Find the prime factorization of each of the following integers.

    1. 39

      39 = 3×13

    2. 85

      85 = 5×17

    3. 101

      101 = 101 (101 is prime.)

  3. List the positive integers that are less than 12 and that are relatively prime to 12.

    1, 5, 7, 11

  4. What are the greatest common divisors of each of the following pairs of integers?

    1. 22×33×55 and 24×34×52

      22×33×52 = 2700

    2. 22×7 and 53×13

      1

  5. Suppose that x and y are two positive integers where xy = 27×38×52×711 and gcd(x, y) = 23×34×5. What is lcm(x, y)?

    lcm(x, y) = xy/gcd(x, y). So the answer is (27×38×52×711)/(23×34×5) = 24×34×5×711 = 12,813,077,294,640

  6. What is gcd(1529, 14039)?

    139

    1. Approximately how many prime numbers are < 1,000,000?

      72382

    2. Approximately how many prime numbers are < 2,000,000?

      137849

    3. Approximately how many prime numbers are < 2,000,000 and > 1,000,000?

      65466. (65466 is not exactly 137849 − 72382 due to roundoff error.)