Determine whether each of the following integers is prime. (Answer yes for prime, no for not prime.)
19
Yes
71
Yes
143
No. (143 = 11×13)
97
Yes
1,113
No. (1113 = 3×7×53)
Find the prime factorization of each of the following integers.
39
39 = 3×13
85
85 = 5×17
101
101 = 101 (101 is prime.)
List the positive integers that are less than 12 and that are relatively prime to 12.
1, 5, 7, 11
What are the greatest common divisors of each of the following pairs of integers?
22×33×55 and 24×34×52
22×33×52 = 2700
22×7 and 53×13
1
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
What is gcd(1529, 14039)?
139
Approximately how many prime numbers are < 1,000,000?
72382
Approximately how many prime numbers are < 2,000,000?
137849
Approximately how many prime numbers are < 2,000,000 and > 1,000,000?
65466. (65466 is not exactly 137849 − 72382 due to roundoff error.)