Write the prime factorization of each number as a product of powers

Write the prime factorization of each number as a product of powers. (a) 129. 3615. 1695 (b) 1613. 107.8112

The correct answer and explanation is :

Prime Factorization of Each Number

(a) 129, 3615, 1695

Prime Factorization of 129:

1. Start by dividing by 3 (since the sum of digits, 1 + 2 + 9 = 12, is divisible by 3):
   [
   129 \div 3 = 43
   ]
2. 43 is a prime number (it is divisible only by 1 and itself).

Thus, the prime factorization of 129 is:
[
129 = 3^1 \times 43^1
]

Prime Factorization of 3615:

1. Check divisibility by 5 (since it ends in 5):
   [
   3615 \div 5 = 723
   ]
2. Now, check divisibility of 723 by 3 (since the sum of digits, 7 + 2 + 3 = 12, is divisible by 3):
   [
   723 \div 3 = 241
   ]
3. Check if 241 is divisible by any prime numbers. It’s not divisible by 2, 3, or 5. It turns out 241 is a prime number.

Thus, the prime factorization of 3615 is:
[
3615 = 5^1 \times 3^1 \times 241^1
]

Prime Factorization of 1695:

1. Since the last digit is 5, we divide by 5:
   [
   1695 \div 5 = 339
   ]
2. Check divisibility of 339 by 3 (since the sum of digits, 3 + 3 + 9 = 15, is divisible by 3):
   [
   339 \div 3 = 113
   ]
3. 113 is a prime number.

Thus, the prime factorization of 1695 is:
[
1695 = 5^1 \times 3^1 \times 113^1
]

---

(b) 1613, 107, 8112

Prime Factorization of 1613:

1. 1613 is not divisible by 2, 3, 5, 7, or 11. Checking further primes, we find 1613 is divisible by 7:
   [
   1613 \div 7 = 230.43
   ]
   So, 1613 is prime.

Thus, the prime factorization of 1613 is:
[
1613 = 1613^1
]

Prime Factorization of 107:

1. Since 107 is not divisible by any primes up to its square root, 107 is a prime number.

Thus, the prime factorization of 107 is:
[
107 = 107^1
]

Prime Factorization of 8112:

1. Since 8112 is even, divide by 2:
   [
   8112 \div 2 = 4056
   ]
   [
   4056 \div 2 = 2028
   ]
   [
   2028 \div 2 = 1014
   ]
   [
   1014 \div 2 = 507
   ]
2. 507 is divisible by 3 (since 5 + 0 + 7 = 12, divisible by 3):
   [
   507 \div 3 = 169
   ]
3. 169 is divisible by 13:
   [
   169 \div 13 = 13
   ]
4. 13 is a prime number.

Thus, the prime factorization of 8112 is:
[
8112 = 2^4 \times 3^1 \times 13^2
]

---

Explanation

Prime factorization is the process of breaking down a composite number into a product of prime numbers. Every integer greater than 1 can be expressed uniquely as a product of primes, except for the order of the factors.

For example:

• 129 is divisible by 3 because the sum of its digits (1 + 2 + 9 = 12) is divisible by 3. After dividing, we get 43, a prime number. Hence, the prime factorization of 129 is ( 3^1 \times 43^1 ).
• 3615 is divisible by 5 because it ends in 5, then by 3 (sum of digits = 12). The result is 241, a prime number.
• Similarly, for 1695, we first divide by 5 (ends in 5) and then by 3, yielding 113, which is prime.

For larger numbers, like 8112, divisibility rules help: start by dividing by the smallest primes (2, 3, 5), and then check divisibility by other primes. This process of successive division reveals the prime factorization of a number.