Classifying whole numbers as prime and composite in different situations

Subsection 1.1.2 Classifying whole numbers as prime and composite in different situations

Whole numbers can be classified as odd, even, prime or composite based on their divisibility and factor properties.

Teacher Resource 1.1.9.

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Online step-by-step guide

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Activity 1.1.2.

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Working in groups, write numbers from \(1\) to \(30\) in a grid with ten numbers on each row, like so:

\begin{equation*} \begin{matrix} 1 \amp 2 \amp 3 \amp 4 \amp 5 \amp 6 \amp 7 \amp 8 \amp 9 \amp 10\\ 11 \amp 12 \amp 13 \amp 14 \amp 15 \amp 16 \amp 17 \amp 18 \amp 19 \amp 20\\ 21 \amp 22 \amp 23 \amp 24 \amp 25 \amp 26 \amp 27 \amp 28 \amp 29 \amp 30 \end{matrix} \end{equation*}

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Draw a circle around all the numbers that are multiples of \(2\) (such as \(4, 6, 8\)), except for \(2\) itself
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Draw a square around all the numbers that are divisible by \(3\) (such as \(6, 9, 12\)), except for \(3\) itself
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Draw a diamond (a tilted square) around all the numbers that are divisible by \(5\) (such as \(10, 15, 20\)), except for \(5\) itself
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What patterns do you notice with the circles, squares, and diamonds?
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Numbers which have no divisors apart from 1 and themselves are called prime. Write down all the numbers which do not have a circle, square, or diamond around them.
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Numbers which do have divisors apart from 1 and themselves are called composite. Write down all the numbers which have a circle, square, or diamond around them.
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Identify prime numbers that are even.
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Brainstorm a real life example where you can prime and composite numbers e.g. sharing seven pens with three people fairly.
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Share your work with your fellow learners.
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Further activity.

Mutula is organizing a party, and he has \(35\) party hats. Can Mutula arrange the hats in rows where each row has the same number of hats? What does this tell you about the number \(35\text{?}\)

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Key Takeaway 1.1.10.

A prime number is a number that has only two factors, that is, \(1\) and itself. Example:\(2,3,5,7,11,\ldots\)

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A factor of a number is a natural number that divides it exactly with no remainder.

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For example, \(6\) is not a prime number because it has more than two factors: \(1, 2, 3,\) and \(6\text{.}\) That is, \(6 = 1 \times 6\) and \(6 = 2 \times 3\text{.}\)

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To identify if a number is prime, check if the number has only two factors: \(1\) and itself.

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Composite numbers are natural numbers greater than \(1\) that have more than two factors.

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Examples: \(4(\text{factors}:\, 1,2,4), 6(\text{factors:} \,1,2,3,6), 9(\text{factors:} \,1,3,9)\text{.}\) Hence \(4,6\) and \(9\) are composite numbers since they have more than \(2\) factors.

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To identify if a number is composite, check if the number has more than two factors. In other words, if it can be divided exactly by numbers other than 1 and itself, then it is a composite number.

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Examples:

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\(4\) is composite because \(4 = 1 \times 4 \) and \(2 \times 2 \text{.}\)
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\(9\) is composite because \(9 = 1 \times 9\) and \(3 \times3\text{.}\)
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\(10\) is composite because \(10 = 1 \times 10\) and \(2 \times 5\)
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Properties

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Every composite number has prime factors. That is, it can be broken down into a product of prime numbers.

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The only even prime number is \(2\) .
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All other even numbers greater than \(2\) are composite.
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The smallest composite number is \(4\text{.}\)
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Odd composite numbers are odd natural numbers greater than \(1\) that are not prime (e.g., \(9, 15, 21,...)\text{.}\)
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Think of prime numbers as building blocks, and composite numbers as being made by combining those blocks.

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Note 1.1.11.

\(0\) and \(1\) are neither prime nor composite.

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Example 1.1.12.

Which of the following numbers are prime and which are composite?

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\(\displaystyle 29\)
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\(\displaystyle 21\)
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\(\displaystyle 5\)
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\(\displaystyle 30\)
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Solution.

\(29\) has no factors other than \(1\) and \(29\) itself.
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Hence \(29\) is a prime number.
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\(21 = 3 \times 7\) meaning it is divisible by both \(3\) and \(7\text{.}\) it is also divisible by \(1\) and itself.
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Therefore, \(21\) is a composite number.
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\(5\) cannot be divided by any number other than \(1\) and \(5\) itself.
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This means that \(5\) is a prime number.
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\(30 = 2 \times 15\) which implies that \(2\) and \(15\) are its factors. Since \(15 = 3 \times 5\text{,}\) these are also factors. Hence \(30\) is divisible by: \(1,2,3,5,10,15,30\text{.}\)
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Therefore, \(30\) is a composite number
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Checkpoint 1.1.13. Classification of Prime and Composite Numbers.

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Checkpoint 1.1.14. Classifying a Number as Odd, Even, Prime or Composite.

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Exercises Exercises

1.

Classify the following numbers as prime or composite.

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\(\displaystyle 14\)
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\(\displaystyle 11 \)
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\(\displaystyle 3\)
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\(\displaystyle 25\)
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\(\displaystyle 17\)
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\(\displaystyle 18\)
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Answer.

\(14\) is composite, as its factors are \(1,2,7,14\) (\(14 = 1 \times 14 = 2 \times 7 = \))
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\(11\) is prime, it has no factors other than \(1\) and itself.
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\(3\) is prime, it has no factors other than \(1\) and itself.
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\(25\) is composite, as its factors are \(1,5,25\text{,}\) (\(25 = 1 \times 25 = 5 \times 5\))
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\(17\) is prime, it has no factors other than \(1\) and itself.
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\(18\) is composite, as its factors are \(1,2,9,18\text{,}\) (\(18 = 1 \times 18 = 2 \times 9\))
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2.

Simplify each of the following expressions and state whether the result is a prime or composite number.

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\(\displaystyle 1024 \times 5 \div 4\)
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\(\displaystyle \sqrt{144} \times 3 - 9 +4\)
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\(\displaystyle \sqrt{64} \times 5\)
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\(\displaystyle 4^2 \times 2 + 4\)
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\(\displaystyle 49^2 + 6 \div 7\)
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\(\displaystyle \sqrt{25} \times 2 - 8\)
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Answer.

\(1024 \times 5 \div 4 = 1280\text{,}\) which is composite.
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\(\sqrt{144} \times 3 - 9 +4 = 31\text{,}\) which is prime.
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\(\sqrt{64} \times 5 = 40\text{,}\) which is composite.
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\(4^2 \times 2 + 4 = 36\text{,}\) which is composite.
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\(49^2 + 6 \div 7 = 2401 + \frac{6}{7} = 2401.857\) when given to 3 decimal places, and is classified as neither prime nor composite.
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\(\sqrt{25} \times 2 - 8 = 2\text{,}\) which is prime.
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3.

The number \(51\) is suspected to be prime. Use divisibility rules to determine whether it is a prime or composite number.

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Answer.

\(51 = 3 \times 17\) Therefore, \(51\) is a composite number.

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4.

A teacher writes a two-digit number on the board. The number is prime, less than \(30\text{,}\) and ends with \(3\text{.}\) List all possible numbers it could be.

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Answer.

The possible two-digit prime numbers that end with \(3\) and are less than \(30\) are: \(13, 23\text{.}\)

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5.

A marathon is divided into \(42\)-kilometer relay sections. Each runner must cover a distance (in km) that is a composite number. List three possible distances a runner could cover.

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Answer.

Any three composite numbers less than or equal to \(42\) km, such as \(4\) km, \(6\) km, and \(9\) km.

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6.

A class of students forms a rectangular grid. The total number of students is \(273\text{.}\) Determine whether this number is prime or composite and explain your reasoning.

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Answer.

\(273 = 13 \times 21\) Therefore, \(273\) is a composite number.

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7.

If a number is divisible by \(2\) and \(3\text{,}\) what smallest composite number could it be?

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Answer.

The smallest composite number divisible by both \(2\) and \(3\) is \(6\text{.}\)

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