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Chapter 5: Problem 43

Find the prime factorization of each composite number. 85,800

Short Answer

Expert verified
The prime factorization of 85,800 is \(2^3 \times 3 \times 5^2 \times 11^1 \times 13^1\).

Step by step solution

01

Understand the concept of Prime Factorization

Prime factorization is finding which prime numbers multiply together to make the original number. Prime numbers are: numbers that have only two factors - one and the number itself.
02

Factorize the number

Let's start by finding the prime factors of 85,800. Start by dividing the number by the smallest prime number, which is 2. 85,800 梅 2 = 42,900. Repeat the process with the result. 42,900 梅 2 = 21,450. Repeat one more time: 21,450 梅 2 = 10,725. Now, the number 10,725 is not divisible by 2, try the next prime number which is 3. It is not divisible by 3 as well. The next prime number, 5, can divide 10,725 to give 2,145. Continue with the prime numbers and 2,145 梅 5 = 429. Keep going, and next, 429 梅 11 = 39 and then 39 梅 3 = 13. At this point, we are left with 13 which is a prime number itself.
03

Write down the Prime Factorization result

So, the prime factorization of 85,800 is: \(2^3 \times 3 \times 5^2 \times 11^1 \times 13^1\). We see the numbers 2, 3, 5, 11, and 13 appearing in the factorization because these are the prime numbers that divide into 85,800.

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