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explain-factors-and-multiples-with-clear-examples

🚀 Factors and multiples are fundamental concepts in mathematics that help us understand how numbers relate to each other. A factor is a number that divides another number without leaving a remainder. For example, if we take the number 12, its factors are 1, 2, 3, 4, 6, and 12 because each of these numbers can divide 12 evenly. On the other hand, a multiple is the result of multiplying a number by an integer. For instance, the multiples of 3 are 3, 6, 9, 12, 15, and so on, because these numbers can be obtained by multiplying 3 by 1, 2, 3, 4, and 5 respectively. Understanding factors and multiples is essential for solving problems related to division, multiplication, and even fractions.

Theory Explanation

Understanding Factors

Factors are numbers that can be multiplied together to get another number. To find the factors of a number, you can start by dividing the number by integers starting from 1 up to that number. If the division results in a whole number (no remainder), then both the divisor and the quotient are factors of the original number.

Understanding Multiples

Multiples of a number are found by multiplying that number by whole numbers (1, 2, 3, ...). For example, to find the multiples of 4, you multiply 4 by 1, 2, 3, and so on, resulting in 4, 8, 12, 16, etc. This means that multiples of a number are infinite, as you can keep multiplying by larger integers.

Finding Common Factors and Multiples

Common factors are factors that two or more numbers share. To find common factors, list the factors of each number and identify the ones that appear in both lists. Common multiples are multiples that two or more numbers share. To find common multiples, list the multiples of each number and identify the ones that appear in both lists.

Key Points

  • 🎯 A factor is a number that divides another number evenly.
  • 🎯 A multiple is the result of multiplying a number by an integer.
  • 🎯 Every number is a multiple of itself and 1.
  • 🎯 Factors come in pairs that multiply to give the original number.
  • 🎯 Finding common factors and multiples is useful in simplifying fractions.

Examples:💡

Find the factors of 18.

Solution:

Step 1: Start dividing 18 by integers starting from 1. 18 ÷ 1 = 18, so 1 and 18 are factors.

\[ 18 \div 1 = 18 \]

Step 2: Next, divide by 2. 18 ÷ 2 = 9, so 2 and 9 are factors.

\[ 18 \div 2 = 9 \]

Step 3: Now divide by 3. 18 ÷ 3 = 6, so 3 and 6 are factors.

\[ 18 \div 3 = 6 \]

Step 4: The factors of 18 are 1, 2, 3, 6, 9, and 18.

Common Mistakes

  • Mistake: Confusing factors with multiples. Students may think that factors are the same as multiples.

    Correction: Remember that factors divide evenly into a number, while multiples are what you get when you multiply a number.

  • Mistake: Forgetting to include 1 and the number itself as factors.

    Correction: Always remember that every number has at least two factors: 1 and itself.

  • Mistake: Not checking for all possible factors up to the number.

    Correction: Make sure to check all integers from 1 up to the number to find all factors.

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