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Chapter 5: Problem 37

Find the prime factorization of each composite number. 1440

Short Answer

Expert verified
The prime factorization of 1440 is \(2^5 * 3^2 * 5^1\).

Step by step solution

01

Define Prime Factorization and Understand the Goal

Prime factorization is the process of determining the prime numbers that multiply together to make a certain number. Here, the goal is to find the prime numbers that multiply together to get the composite number 1440.
02

Start with the Smallest Prime Number

Begin the factorization process with the smallest prime number, which is 2. By checking if 1440 can be divided evenly by 2, it's found that it can, giving a quotient of 720.
03

Continue Factoring

Repeat the process using the quotient from the previous step. Divide 720 by 2 again, yielding 360. Keep repeating the process until the quotient is no longer divisible by 2. This occurs when getting a quotient of 45, which is not divisible by 2.
04

Move to the Next Prime Number

Move to the next prime number, which is 3. 45 is divisible by 3, yielding a quotient of 15. Continue the process to get a quotient of 5, which, being a prime number itself, is the final step in the prime factorization.
05

Write Down the Prime Factorization

Combine all the prime numbers used to divide the original number, in their correct multiples, to give the prime factorization. Here, the prime factorization is \(2^5 * 3^2 * 5^1\).

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