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Factors and Multiples - Prime Factorization

Grade 5ICSE

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

🔑Concepts

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Factors are numbers that divide another number exactly without leaving any remainder. Multiples are the products of a given number and any whole number. Visually, if you have 15 squares, you can arrange them in a 3×53 \times 5 rectangle, which shows that 33 and 55 are factors of 1515.

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Prime Numbers are numbers greater than 11 that have exactly two factors: 11 and the number itself (e.g., 2,3,5,7,112, 3, 5, 7, 11). Composite Numbers have more than two factors. Visually, imagine a 'Prime Sieve' where you cross out even numbers (except 22) and multiples of other primes to reveal the prime building blocks of all numbers.

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Prime Factorization is the process of expressing a composite number as the product of its prime factors. Every composite number can be uniquely broken down into these prime 'atoms'. For example, the number 1212 is always 2×2×32 \times 2 \times 3.

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The Factor Tree Method is a visual branching diagram used to find prime factors. You write the number at the top and draw two 'branches' for any factor pair. You continue branching until every end-point is a prime number. Visually, you circle these prime 'leaves' at the ends of the branches to collect your final answer.

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The Division Method (or Ladder Method) involves repeated division by the smallest possible prime numbers. You draw a vertical line with the number on the right and the prime divisor on the left. The quotient is written below the number, creating a ladder-like structure that continues until the quotient reaches 11.

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Index Notation is a shorthand way to write repeated prime factors using exponents. If a factor like 22 appears three times, we write it as 232^3. Visually, the base is the prime factor and the small superscript number (exponent) indicates the count of that factor in the product.

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The number 11 is unique because it is neither prime nor composite. It has only one factor (11 itself), whereas prime numbers must have exactly two distinct factors. In visual diagrams, 11 is never used as a branch in a factor tree or a divisor in the ladder method.

📐Formulae

Composite Number=P1×P2×P3×…\text{Composite Number} = P_1 \times P_2 \times P_3 \times \dots (where PP represents prime factors)

a^n = a \times a \times a \dots \text{ (n times)}

Example: 23=2×2×2=8\text{Example: } 2^3 = 2 \times 2 \times 2 = 8

💡Examples

Problem 1:

Find the prime factorization of 7272 using the Factor Tree method and express it in index notation.

Solution:

Step 1: Start with 7272. Split it into 2×362 \times 36. Circle the prime number 22. \nStep 2: Split 3636 into 2×182 \times 18. Circle the prime number 22. \nStep 3: Split 1818 into 2×92 \times 9. Circle the prime number 22. \nStep 4: Split 99 into 3×33 \times 3. Circle both prime numbers 33 and 33. \nStep 5: Collect all circled primes: 2×2×2×3×32 \times 2 \times 2 \times 3 \times 3. \nStep 6: Write in index notation: 23×322^3 \times 3^2.

Explanation:

The factor tree helps visualize the breakdown of 7272. We keep dividing until only prime numbers remain at the ends of the branches.

Problem 2:

Find the prime factorization of 120120 using the Division Method.

Solution:

Step 1: Divide 120120 by the smallest prime 22: 120÷2=60120 \div 2 = 60. \nStep 2: Divide 6060 by 22: 60÷2=3060 \div 2 = 30. \nStep 3: Divide 3030 by 22: 30÷2=1530 \div 2 = 15. \nStep 4: Divide 1515 by the next smallest prime 33: 15÷3=515 \div 3 = 5. \nStep 5: Divide 55 by the prime 55: 5÷5=15 \div 5 = 1. \nStep 6: The prime factors are the divisors: 2×2×2×3×5=23×3×52 \times 2 \times 2 \times 3 \times 5 = 2^3 \times 3 \times 5.

Explanation:

In the division method, we perform successive divisions by prime numbers until the quotient becomes 11. The product of these divisors gives the prime factorization.