Let's go step-by-step through the questions:

1. Determine whether each of these integers is prime.

A prime number is a number greater than 1 that has no positive divisors other than 1 and itself.

• a) 21
  21 is not prime because it is divisible by 3 and 7 (21 = 3 × 7).

• b) 29
  29 is prime because it has no divisors other than 1 and 29.

• c) 71
  71 is prime because it has no divisors other than 1 and 71.

• d) 97
  97 is prime because it has no divisors other than 1 and 97.

• e) 111
  111 is not prime because it is divisible by 3 (111 = 3 × 37).

• f) 143
  143 is not prime because it is divisible by 11 and 13 (143 = 11 × 13).

2. Determine whether each of these integers is prime.

• a) 19
  19 is prime because it has no divisors other than 1 and 19.

• b) 27
  27 is not prime because it is divisible by 3 (27 = 3 × 9).

• c) 93
  93 is not prime because it is divisible by 3 and 31 (93 = 3 × 31).

• d) 101
  101 is prime because it has no divisors other than 1 and 101.

• e) 107
  107 is prime because it has no divisors other than 1 and 107.

• f) 113
  113 is prime because it has no divisors other than 1 and 113.

3. Find the prime factorization of each of these integers.

The prime factorization of a number is the expression of that number as a product of prime numbers.

• a) 88
  $88 = 2 \times 2 \times 2 \times 11 = 2^3 \times 11$.

• b) 126
  $126 = 2 \times 3 \times 3 \times 7 = 2 \times 3^2 \times 7$.

• c) 729
  $729 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 3^6$.

• d) 1001
  $1001 = 7 \times 11 \times 13$.

• e) 1111
  $1111 = 11 \times 101$ (both primes).

• f) 909,090
  $909,090 = 2 \times 3 \times 3 \times 5 \times 7 \times 7 \times 11 = 2 \times 3^2 \times 5 \times 7^2 \times 11$.

4. Find the prime factorization of each of these integers.

• a) 39
  $39 = 3 \times 13$.

• b) 81
  $81 = 3 \times 3 \times 3 \times 3 = 3^4$.

• c) 101
  101 is prime, so the factorization is just $101$.

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Would you like further details or explanations on any of these steps? If you have any questions, feel free to ask!

Related Questions:

1. How can you efficiently test if a large number is prime?
2. What are some methods for factoring large integers quickly?
3. How does the Sieve of Eratosthenes help in finding prime numbers?
4. Why are prime numbers important in cryptography?
5. Can a prime number have more than two divisors? Why or why not?

Tip: When checking if a number is prime, you only need to test divisibility up to the square root of the number, making the process faster for larger numbers.