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Chapter 1: Problem 16

Identify each natural number as prime or composite. If the number is composite, find its prime factorization. $$75$$

Short Answer

Expert verified
The number 75 is a composite number. Its prime factorization is \(75 = 3 \times 5 \times 5\) or \(75 = 3 \times 5^2\).

Step by step solution

01

Identifying the number

It is necessary to identify whether 75 is a prime or composite number. A prime number has only two distinct divisors: 1 and itself. But 75 has more than two divisors which are 1, 3, 5, 15, 25, and 75. Therefore, 75 is a composite number.
02

Finding prime factors

The next step is to find the prime factors of 75. To do that, start dividing 75 by the smallest prime number, which is 2. Since 75 is not divisible by 2, try the next prime number which is 3. 75 is divisible by 3 which gives the result 25. Continue this process with 25, dividing it by prime numbers. The smallest prime number that divides 25 is 5. Hence, 25 is 5*5. Thus, the prime factors of 75 are 3, 5, and 5.
03

Writing the prime factorization

Now that the prime factors have been found, the prime factorization can be written as a product of prime factors. The prime factors of 75 are 3, 5, and 5. Hence, the prime factorization are written as \(75 = 3 \times 5 \times 5\) or \(75 = 3 \times 5^2\).

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