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Number Sense and Operations - Prime and Composite Numbers

Grade 5IB

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Prime Numbers: A prime number is a whole number greater than 11 that has exactly two factors: 11 and the number itself. Visual concept: If you have a prime number of blocks, such as 77, the only way to arrange them into a perfect rectangle is a single row of 1×71 \times 7 or a single column of 7×17 \times 1.

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Composite Numbers: A composite number is a whole number greater than 11 that has more than two factors. Visual concept: Unlike prime numbers, composite numbers can be arranged into multiple different rectangular shapes. For example, 1212 blocks can be arranged as 1×121 \times 12, 2×62 \times 6, or 3×43 \times 4.

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The Special Case of 0 and 1: The number 11 is neither prime nor composite because it has only one factor (itself). The number 00 is also neither prime nor composite as it has an infinite number of factors and is not greater than 11.

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Factor Trees: This is a tool used to break down a composite number into its prime factors. Visual concept: It starts with the composite number at the top (the 'root') and splits into two 'branches' representing factors. This continue until every branch ends in a prime number (the 'leaves').

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The Sieve of Eratosthenes: An ancient method to identify prime numbers in a sequence. Visual concept: Imagine a grid of numbers from 11 to 100100. By systematically crossing out multiples of 2,3,5,2, 3, 5, and 77, the remaining numbers that are not crossed out are all the prime numbers in that range.

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Prime Factorization: Every composite number can be written as a unique product of prime numbers. Visual concept: This is like finding the 'DNA' or 'building blocks' of a number. For example, the number 3030 is always composed of the primes 2×3×52 \times 3 \times 5.

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Divisibility Rules: These are shortcuts to determine if a number is composite without doing long division. For example, if a number is even (ends in 0,2,4,6,0, 2, 4, 6, or 88), it is divisible by 22 and is composite (except for the number 22 itself).

📐Formulae

Factors of a Prime Number P={1,P}\text{Factors of a Prime Number } P = \{1, P\}

Number of Factors of a Composite Number C>2\text{Number of Factors of a Composite Number } C > 2

Prime Factorization Form: n=p1a×p2b×p3c…\text{Prime Factorization Form: } n = p_{1}^{a} \times p_{2}^{b} \times p_{3}^{c} \dots

Example of Prime Factorization: 24=2×2×2×3=23×3\text{Example of Prime Factorization: } 24 = 2 \times 2 \times 2 \times 3 = 2^{3} \times 3

💡Examples

Problem 1:

Identify whether 2121 is a prime or composite number by listing its factors.

Solution:

Step 1: Find all pairs of numbers that multiply to give 2121. \ Step 2: 1×21=211 \times 21 = 21 and 3×7=213 \times 7 = 21. \ Step 3: List the unique factors: {1,3,7,21}\{1, 3, 7, 21\}. \ Step 4: Count the factors. There are 44 factors. \ Step 5: Since 4>24 > 2, 2121 is a composite number.

Explanation:

A prime number must have exactly two factors. Because 2121 can be divided evenly by 33 and 77 in addition to 11 and 2121, it is composite.

Problem 2:

Find the prime factorization of 6060 using a factor tree.

Solution:

Step 1: Choose any two factors of 6060, such as 66 and 1010. \ Step 2: Break down 66 into 2×32 \times 3. Both 22 and 33 are prime. \ Step 3: Break down 1010 into 2×52 \times 5. Both 22 and 55 are prime. \ Step 4: Collect all the prime numbers at the ends of the branches: 2,3,2,52, 3, 2, 5. \ Step 5: Write the product: 2×2×3×52 \times 2 \times 3 \times 5 or 22×3×52^{2} \times 3 \times 5.

Explanation:

The factor tree method breaks a number into smaller and smaller pieces until only prime numbers remain. Multiplying these primes back together results in the original number.