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Chapter 1: Problem 73

Factor the given number into its prime factors. If the number is prime, say so. $$87$$

Short Answer

Expert verified
The prime factorization of 87 is 3 脳 29.

Step by step solution

01

Check if the number is prime

To determine whether 87 is a prime number, divide it by smaller prime numbers like 2, 3, 5, 7, and 11. If 87 is not divisible by any of these numbers, it is prime. Otherwise, it is not.
02

Check divisibility by 2

Since 87 is an odd number, it is not divisible by 2. Move on to the next smallest prime number, which is 3.
03

Check divisibility by 3

To see if 87 is divisible by 3, add the digits of 87 (i.e., 8 + 7 = 15). Since 15 is divisible by 3, so is 87. Perform the division: \[87 梅 3 = 29\]
04

Check if 29 is a prime number

To confirm if 29 is a prime number, check its divisibility by smaller prime numbers (2, 3, 5). Since 29 is not divisible by any of these, it is a prime number.
05

Write down the factors

Since 87 = 3 脳 29 and both 3 and 29 are prime, the prime factorization of 87 is complete.

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Key Concepts

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

prime numbers
Prime numbers play a vital role in factorization. A prime number is a number greater than 1 that has no positive divisors other than 1 and itself. For example, 2, 3, 5, 7, 11, and 29 are prime numbers. Using prime numbers, we can break down composite numbers (numbers that have more than two factors) into their basic building blocks. For instance, in our problem, 87 can be broken down into the prime factors 3 and 29. Understanding prime numbers helps us simplify many mathematical operations and solve problems efficiently.
divisibility rules
Divisibility rules make it easier to determine if one number can be divided by another without leaving a remainder. For instance, you can quickly tell if a number is divisible by 2 if it ends in an even number (0, 2, 4, 6, 8). If the sum of its digits is divisible by 3, the number itself is divisible by 3, like how 87 is divisible by 3 because 8 + 7 = 15, and 15 is divisible by 3. Similarly, there are other rules for divisibility by 5, 7, 11, and so on. Applying these rules helps them check the divisibility of numbers quickly, as we did for 87.
basic arithmetic operations
Basic arithmetic operations include addition, subtraction, multiplication, and division. These operations form the foundation for solving more complex mathematical problems. For instance, when performing prime factorization of 87:
  • We first checked divisibility by adding digits (8 + 7) and finding the sum (15).
  • Then, we divided 87 by 3 to get 29.
Understanding these operations is essential for tasks like breaking down numbers and confirming their factors. Practicing these basics ensures that you can handle more advanced math concepts easily.

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