To find the whole numbers that are factors of 1260, we first need to determine its prime factorization.
- Find the prime factorization of 1260:
- Divide by 2: \( 1260 \div 2 = 630 \)
- Divide by 2: \( 630 \div 2 = 315 \)
- Divide by 3: \( 315 \div 3 = 105 \)
- Divide by 3: \( 105 \div 3 = 35 \)
- Divide by 5: \( 35 \div 5 = 7 \)
- Divide by 7: \( 7 \div 7 = 1 \)
So, the prime factorization of 1260 is: \[ 1260 = 2^2 \times 3^2 \times 5^1 \times 7^1 \]
- Find all the factors: To find the factors, we take all combinations of the prime factors, including their powers.
The factors of a number can be calculated by taking the exponents in the prime factorization, adding one to each exponent, and then multiplying the results:
- For \( 2^2 \): The choices for the exponent of 2 are \( 0, 1, 2 \) (3 options).
- For \( 3^2 \): The choices for the exponent of 3 are \( 0, 1, 2 \) (3 options).
- For \( 5^1 \): The choices for the exponent of 5 are \( 0, 1 \) (2 options).
- For \( 7^1 \): The choices for the exponent of 7 are \( 0, 1 \) (2 options).
Calculating the total number of factors: \[ (2 + 1) \times (2 + 1) \times (1 + 1) \times (1 + 1) = 3 \times 3 \times 2 \times 2 = 36 \]
Now we can list the factors of 1260. The factors can be generated by taking products of the primes.
The complete list of factors of 1260 is: 1, 2, 3, 4, 5, 6, 7, 9, 10, 12, 14, 15, 18, 20, 21, 28, 30, 36, 42, 60, 63, 70, 84, 90, 105, 126, 140, 180, 252, 315, 420, 630, 1260
So the whole numbers that are factors of 1260 are: 1, 2, 3, 4, 5, 6, 7, 9, 10, 12, 14, 15, 18, 20, 21, 28, 30, 36, 42, 60, 63, 70, 84, 90, 105, 126, 140, 180, 252, 315, 420, 630, 1260.
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