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Chapter 5: Problem 90

List all positive integer factors of each number. $$ 36 $$

Short Answer

Expert verified
1, 2, 3, 4, 6, 9, 12, 18, 36

Step by step solution

01

Understand Factors

Factors of a number are the integers that you can multiply together to get that number. For example, factors of 36 are the numbers that multiply to give 36.
02

Find All Factor Pairs

Begin by finding pairs of numbers that multiply together to equal 36. Start from 1 and work up: 1. 1 and 36 (since \(1 \times 36 = 36\))2. 2 and 18 (since \(2 \times 18 = 36\))3. 3 and 12 (since \(3 \times 12 = 36\))4. 4 and 9 (since \(4 \times 9 = 36\))5. 6 and 6 (since \(6 \times 6 = 36\))
03

List All Unique Factors in Ascending Order

Now that all factor pairs are found, list each unique factor in ascending order: 1, 2, 3, 4, 6, 9, 12, 18, 36.

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Key Concepts

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

positive integer factors
When we talk about factors of a number, we are referring to the integers that can be multiplied together to produce that number. Specifically, positive integer factors are those factors that are positive whole numbers. For example, the positive integer factors of 36 are the numbers that multiply to give 36, such as 1, 2, 3, and so on.
Finding these factors is an essential skill in mathematics and helps in various problem-solving scenarios. Here鈥檚 how you can find the positive integer factors of a number:
  • Always start with the number 1, since 1 is a factor of every number.
  • Identify pairs of numbers that multiply to give the target number.
  • Ensure all factors listed are positive integers.
multiplication pairs
Finding multiplication pairs is a systematic way to determine all factors of a given number. For instance, when finding the factors of 36, you look for pairs of numbers that, when multiplied, equal 36. These pairs are known as multiplication pairs.
Here's a simple way to find them:
  • Start with the number 1 and pair it with 36 (since \(1 \times 36 = 36\).
  • Move to the next number, which is 2, and find its pair (since \(2 \times 18 = 36\).
  • Continue this process until the pairs start to repeat or until you reach the middle factor. In this case, 6 \(6 \times 6 = 36\).
By listing these pairs, you will identify all possible factors. The multiplication pairs for 36 are:
  • 1 and 36
  • 2 and 18
  • 3 and 12
  • 4 and 9
  • 6 and 6
finding factors
Finding factors involves identifying every integer that divides a given number without leaving a remainder. This is crucial for solving many mathematical problems and can be done by following a systematic approach.
Here鈥檚 a step-by-step method to find the factors of 36:
  • Identify the smallest factor, which is always 1.
  • Proceed to check if each successive integer divides 36 evenly. For 36, the first few factors you check would be: 2, 3, 4, etc.
  • If a number divides 36 without a remainder, it is a factor. For example, \(36 \div 2 = 18\) which means 2 and 18 are factors.
  • Continue this until you reach an already-found factor pair or the square root of the number. For 36, \(\sqrt{36} = 6\), which tells you the last factor to check is 6.

Once you find all factors, list them in ascending order. The complete set of positive integer factors of 36 is:
  • 1, 2, 3, 4, 6, 9, 12, 18, 36

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