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Number - Prime Factors, HCF, and LCM

Grade 8IGCSE

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

🔑Concepts

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Prime Number: A number greater than 1 that has only two factors: 1 and itself.

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Prime Factorization: Expressing a composite number as a product of its prime factors (usually in index notation).

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Highest Common Factor (HCF): The largest number that divides exactly into two or more numbers.

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Lowest Common Multiple (LCM): The smallest number that is a multiple of two or more numbers.

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Venn Diagram Method: A visual way to find HCF and LCM by placing common prime factors in the intersection.

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Index Notation: Writing repeated prime factors using powers (e.g., 2×2×2=232 \times 2 \times 2 = 2^3).

📐Formulae

HCF×LCM=A×B\text{HCF} \times \text{LCM} = A \times B (for two numbers AA and BB)

HCF=Product of the lowest powers of common prime factors\text{HCF} = \text{Product of the lowest powers of common prime factors}

LCM=Product of the highest powers of all prime factors present\text{LCM} = \text{Product of the highest powers of all prime factors present}

💡Examples

Problem 1:

Express 120 as a product of its prime factors in index form.

Solution:

120=23×3×5120 = 2^3 \times 3 \times 5

Explanation:

Using a factor tree or repeated division: 120÷2=60120 \div 2 = 60; 60÷2=3060 \div 2 = 30; 30÷2=1530 \div 2 = 15; 15÷3=515 \div 3 = 5; 5÷5=15 \div 5 = 1. The factors are 2,2,2,3,52, 2, 2, 3, 5.

Problem 2:

Find the HCF and LCM of 36 and 48 using prime factorization.

Solution:

HCF = 12, LCM = 144

Explanation:

Prime factors of 36=22×3236 = 2^2 \times 3^2. Prime factors of 48=24×3148 = 2^4 \times 3^1. For HCF, take the lowest powers of common factors: 22×31=4×3=122^2 \times 3^1 = 4 \times 3 = 12. For LCM, take the highest powers of all factors: 24×32=16×9=1442^4 \times 3^2 = 16 \times 9 = 144.

Problem 3:

A blue light flashes every 20 seconds and a red light flashes every 30 seconds. If they both flash at the same time now, after how many seconds will they next flash together?

Solution:

60 seconds

Explanation:

This problem requires finding the LCM of 20 and 30. Multiples of 20: 20, 40, 60, 80... Multiples of 30: 30, 60, 90... The smallest common multiple is 60.