Question
Find all the factors of 144
Answers
GPT-4o mini
To find all the factors of 144, we can start by determining its prime factorization.
1. **Finding the Prime Factorization:**
- Divide by the smallest prime number (2):
- \( 144 ÷ 2 = 72 \)
- \( 72 ÷ 2 = 36 \)
- \( 36 ÷ 2 = 18 \)
- \( 18 ÷ 2 = 9 \)
- Now, divide by the next smallest prime number (3):
- \( 9 ÷ 3 = 3 \)
- \( 3 ÷ 3 = 1 \)
So, the prime factorization of 144 is:
\[
144 = 2^4 \times 3^2
\]
2. **Finding the Factors:**
To find the factors of 144, we can generate all combinations of the prime factors raised to their respective powers.
The exponents for the prime factorization \(2^4 \times 3^2\) allow us to create factors by taking powers of the primes from 0 up to the maximum exponents in the factorization:
- For \(2^a\), \(a\) can be 0, 1, 2, 3, or 4 (5 options).
- For \(3^b\), \(b\) can be 0, 1, or 2 (3 options).
The total number of factors = (number of options for a) × (number of options for b) = \(5 \times 3 = 15\).
3. **Listing the Factors:**
Now we can list the factors by systematically combining the powers:
- \( 2^0 \times 3^0 = 1 \)
- \( 2^0 \times 3^1 = 3 \)
- \( 2^0 \times 3^2 = 9 \)
- \( 2^1 \times 3^0 = 2 \)
- \( 2^1 \times 3^1 = 6 \)
- \( 2^1 \times 3^2 = 18 \)
- \( 2^2 \times 3^0 = 4 \)
- \( 2^2 \times 3^1 = 12 \)
- \( 2^2 \times 3^2 = 36 \)
- \( 2^3 \times 3^0 = 8 \)
- \( 2^3 \times 3^1 = 24 \)
- \( 2^3 \times 3^2 = 72 \)
- \( 2^4 \times 3^0 = 16 \)
- \( 2^4 \times 3^1 = 48 \)
- \( 2^4 \times 3^2 = 144 \)
Putting all together, the factors of 144 are:
\[
1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144
\]
So, the complete list of factors of 144 is:
**1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144.**
1. **Finding the Prime Factorization:**
- Divide by the smallest prime number (2):
- \( 144 ÷ 2 = 72 \)
- \( 72 ÷ 2 = 36 \)
- \( 36 ÷ 2 = 18 \)
- \( 18 ÷ 2 = 9 \)
- Now, divide by the next smallest prime number (3):
- \( 9 ÷ 3 = 3 \)
- \( 3 ÷ 3 = 1 \)
So, the prime factorization of 144 is:
\[
144 = 2^4 \times 3^2
\]
2. **Finding the Factors:**
To find the factors of 144, we can generate all combinations of the prime factors raised to their respective powers.
The exponents for the prime factorization \(2^4 \times 3^2\) allow us to create factors by taking powers of the primes from 0 up to the maximum exponents in the factorization:
- For \(2^a\), \(a\) can be 0, 1, 2, 3, or 4 (5 options).
- For \(3^b\), \(b\) can be 0, 1, or 2 (3 options).
The total number of factors = (number of options for a) × (number of options for b) = \(5 \times 3 = 15\).
3. **Listing the Factors:**
Now we can list the factors by systematically combining the powers:
- \( 2^0 \times 3^0 = 1 \)
- \( 2^0 \times 3^1 = 3 \)
- \( 2^0 \times 3^2 = 9 \)
- \( 2^1 \times 3^0 = 2 \)
- \( 2^1 \times 3^1 = 6 \)
- \( 2^1 \times 3^2 = 18 \)
- \( 2^2 \times 3^0 = 4 \)
- \( 2^2 \times 3^1 = 12 \)
- \( 2^2 \times 3^2 = 36 \)
- \( 2^3 \times 3^0 = 8 \)
- \( 2^3 \times 3^1 = 24 \)
- \( 2^3 \times 3^2 = 72 \)
- \( 2^4 \times 3^0 = 16 \)
- \( 2^4 \times 3^1 = 48 \)
- \( 2^4 \times 3^2 = 144 \)
Putting all together, the factors of 144 are:
\[
1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144
\]
So, the complete list of factors of 144 is:
**1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144.**