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Chapter 3: Problem 78

Write 83 as a product of prime factors. Write 83 as a product of prime factors.

Short Answer

Expert verified
83 = 83

Step by step solution

01

- Understanding Prime Factors

Prime factors are the prime numbers that multiply together to give the original number. So, for the exercise, identify any prime numbers that can multiply together to result in 83.
02

- Check if 83 is Prime

Determining if 83 is a prime number will help us decide if it can be factored further. A prime number has only two divisors: 1 and itself. Check if 83 can be divided by any prime numbers less than its square root, approximately 9.1.
03

- Test for Division by Small Prime Numbers

Test (83 梅 2), (83 梅 3), (83 梅 5), and (83 梅 7). None of these divisions result in an integer, confirming 83 cannot be divided by these primes and is thus prime itself.
04

- Confirming 83 is Prime

Since 83 only divides by 1 and 83 without any remainder, it is confirmed as a prime number.
05

- Express as Product of Prime Factors

As 83 is a prime number, it can only be expressed as the product of itself.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Prime Numbers
Prime numbers are fundamental in math. They are numbers greater than 1 that have no divisors other than 1 and themselves. Examples include 2, 3, 5, 7, and 11.
To identify if a number is prime, you check if it has any divisors other than 1 and itself. If it doesn't, it's prime.
For instance, 2 is a prime because it can only be divided by 1 and 2. However, 6 is not prime because it can be divided by 1, 2, 3, and 6.
Factorization
Factorization is breaking down a number into its components that, when multiplied together, give the original number. For example, the factors of 6 are 2 and 3 because 2 * 3 = 6.
Prime factorization is similar but involves breaking down the number into prime numbers only. Let's take the number 20. Its prime factors are 2, 2, and 5, because 2 * 2 * 5 = 20.
In the given exercise, we need to express 83 as a product of prime factors. This means determining whether 83 can be broken down further, which involves checking smaller prime numbers.
Divisibility Rules
Divisibility rules help us quickly determine if a number can be divided evenly by another number without performing full division.
Here are some useful divisibility rules:
  • A number is divisible by 2 if its last digit is even.
  • A number is divisible by 3 if the sum of its digits is divisible by 3.
  • A number is divisible by 5 if its last digit is 0 or 5.
  • A number is divisible by 7, 11, or 13 through more complex rules often involving division tricks.
In the case of 83, we tested divisibility by 2, 3, 5, and 7 and found it wasn't divisible by any, confirming it as a prime number.

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