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Number - Highest Common Factor (HCF) and Lowest Common Multiple (LCM)

Grade 6IB

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

🔑Concepts

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Factors and Multiples: Factors are whole numbers that divide into another number exactly without leaving a remainder. If you visualize a number like 1212 as a set of tiles, the factors are the possible side lengths of a rectangle formed by those tiles (1×12,2×6,3×41 \times 12, 2 \times 6, 3 \times 4). Multiples are the result of multiplying a number by an integer. You can visualize multiples as equal-sized jumps along a number line, starting from zero.

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Prime and Composite Numbers: A prime number has exactly two factors: 11 and itself. A composite number has more than two factors. Visually, prime numbers (except 22) are always odd, and in an array or grid, they cannot be arranged into a perfect rectangle other than a single row or column.

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Prime Factorization: Every composite number can be broken down into a product of prime numbers. This is often represented visually using a 'Factor Tree,' where the original number sits at the top and branches out into pairs of factors until every branch ends in a prime number 'leaf'.

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Index Notation: To simplify prime factorization, we use powers (indices) for repeated factors. For example, 2×2×2×52 \times 2 \times 2 \times 5 is written as 23×52^{3} \times 5. This notation makes it easier to compare the 'building blocks' of different numbers.

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Highest Common Factor (HCF): The HCF is the largest factor shared by two or more numbers. If you place the prime factors of two numbers into a Venn Diagram, the HCF is found by multiplying the numbers located in the overlapping central section where the circles intersect.

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Lowest Common Multiple (LCM): The LCM is the smallest number that is a multiple of two or more numbers. In a Venn Diagram of prime factors, the LCM is the product of every single number shown in the entire diagram (the union of the circles).

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Co-prime Numbers: Two numbers are said to be co-prime if their only common factor is 11. In this case, their HCF is 11 and their LCM is simply the product of the two numbers. Visually, their Venn Diagram circles would have no prime factors in the overlapping intersection.

📐Formulae

HCF(a,b)×LCM(a,b)=a×bHCF(a, b) \times LCM(a, b) = a \times b

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

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

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

💡Examples

Problem 1:

Find the HCF and LCM of 3636 and 4848 using prime factorization.

Solution:

Step 1: Find the prime factors of each number. 36=2×18=2×2×9=2×2×3×3=22×3236 = 2 \times 18 = 2 \times 2 \times 9 = 2 \times 2 \times 3 \times 3 = 2^{2} \times 3^{2} 48=2×24=2×2×12=2×2×2×6=2×2×2×2×3=24×3148 = 2 \times 24 = 2 \times 2 \times 12 = 2 \times 2 \times 2 \times 6 = 2 \times 2 \times 2 \times 2 \times 3 = 2^{4} \times 3^{1}

Step 2: To find the HCF, take the lowest power of the common prime factors. HCF=22×31=4×3=12HCF = 2^{2} \times 3^{1} = 4 \times 3 = 12

Step 3: To find the LCM, take the highest power of all prime factors present. LCM=24×32=16×9=144LCM = 2^{4} \times 3^{2} = 16 \times 9 = 144

Explanation:

We decompose both numbers into their prime building blocks. For the HCF, we look for what they share (the lowest powers). For the LCM, we ensure every prime factor is included at its maximum frequency to cover both numbers.

Problem 2:

A bus to City A leaves every 1515 minutes, and a bus to City B leaves every 2020 minutes. If both buses leave at 9:00 AM, at what time will they next leave together?

Solution:

Step 1: Identify that we need the Lowest Common Multiple (LCM) of 1515 and 2020 to find the first time their schedules align. Step 2: Find prime factors. 15=3×515 = 3 \times 5 20=2×2×5=22×520 = 2 \times 2 \times 5 = 2^{2} \times 5 Step 3: Calculate the LCM. LCM=22×3×5=4×3×5=60LCM = 2^{2} \times 3 \times 5 = 4 \times 3 \times 5 = 60 minutes. Step 4: Add the LCM to the starting time. 9:00 AM+60 minutes=10:00 AM9:00\text{ AM} + 60\text{ minutes} = 10:00\text{ AM}.

Explanation:

Since this is a repeating cycle problem, the LCM helps us find the first point in the future where both cycles complete at the exact same moment.