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      Expressing an integer as a product of its prime factors

      Lesson details

      Learning outcome

      I can write any positive integer uniquely as a product of its prime factors.

      Key learning points

      1. Every composite number has prime number factors.
      2. All the factors of a number can be broken down into prime factors.
      3. Every positive integer can be broken down into prime factors.
      4. Every positive integer can be written as a product of its prime factors.
      5. Prime factor products can be simplified by using index notation.

      Keywords

      • Composite - A composite number is an integer with more than two factors. All integers greater than 1 are either composite or prime

      • Prime factors - Prime factors are the factors of a number that are, themselves, prime.

      • Unique - Each composite number has a unique product of its prime factors. i.e. there is only one unique way it can be written.

      Common misconception

      Use of addition instead of multiplication when the decomposing the number.

      Reiterate that the product of prime factors is unique.

      Teacher tip

      Get students to check their solutions on a calculator.

      Licence

      This content is © Oak National Academy Limited (2025), licensed on Open Government Licence version 3.0
      except where otherwise stated. See Oak's terms & conditions
      (Collection 2).

      Lesson video

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      Prior knowledge starter quiz

      6 Questions

      Q1.
      Fill in the missing word in this statement: A number which has exactly 2 factors is called a number.

      Correct Answer: prime

      Q2.
      Which of the following arrays shows that 6 is not a prime number?

      An image in a quiz
      An image in a quiz
      Correct Answer: An image in a quiz
      An image in a quiz

      Q3.
      Which of the following is not a prime number?

      11
      Correct answer: 15
      17
      19

      Q4.
      True or false? 31 is a prime number.

      Correct answer: True, it has exactly 2 factors, 1 × 31
      False, you can make an array of 3 × 11

      Q5.
      True or false? All odd numbers between 2 and 10 are prime numbers.

      True, you cannot make an array with a width of 2
      Correct answer: False, 9 has 3 factors

      Q6.
      Which is the odd one out in this list? 37, 47, 57, 67

      37
      47
      Correct answer: 57
      67

      6 Questions

      Q1.
      Fill in the missing word: Integers greater than 1 are either or prime.

      Correct Answer: composite

      Q2.
      What number is missing in this statement? Composite numbers have more than factors

      1
      Correct answer: 2
      3
      4

      Q3.
      What is the missing number in the first step in the process to find prime factors of 12? $$12 = 2 ×$$

      Correct Answer: 6, six

      Q4.
      What is the missing product in this process to find prime factors of 12?

      An image in a quiz
      Correct answer: 2 × 3
      2 + 3
      3 + 2
      1 × 6
      3 + 3

      Q5.
      Given that 120 = $$2^3\times3\times5$$, which of the following equals 600?

      $$2^4\times3\times5$$
      Correct answer: $$2^3\times3\times5^2$$
      $$2^3\times3^2\times5$$

      Q6.
      Given that 3600 = $$2^4\times3^2\times5^2$$, which of the following equals 360?

      Correct answer: $$2^3\times3^2\times5$$
      $$2^2\times3\times5$$
      $$2^5\times3^2\times5^3$$

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