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Number System - Playing with Numbers: Factors, Multiples, HCF, LCM

Grade 6ICSE

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

🔑Concepts

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Factors and Multiples: A factor is an exact divisor of a number, while a multiple is a number obtained by multiplying it by any natural number. For example, the factors of 1212 are 1,2,3,4,6,121, 2, 3, 4, 6, 12. Visually, factors can be understood by arranging 1212 blocks into perfect rectangles, such as a 3×43 \times 4 or 2×62 \times 6 grid.

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Prime and Composite Numbers: A prime number is a natural number greater than 11 that has exactly two factors, 11 and the number itself (e.g., 2,3,5,7,112, 3, 5, 7, 11). A composite number has more than two factors. 11 is a unique number that is neither prime nor composite. You can visualize prime numbers using the Sieve of Eratosthenes, a grid where you cross out all multiples to leave only the primes.

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Divisibility Rules: These are mental math shortcuts to determine if a number is divisible by another without full division. A number is divisible by 33 if the sum of its digits is divisible by 33; it is divisible by 44 if its last two digits are divisible by 44; and it is divisible by 66 if it is divisible by both 22 (even) and 33.

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Prime Factorization: Every composite number can be expressed as a unique product of prime numbers. This is often represented using a 'Factor Tree', which looks like an upside-down tree where the number at the top splits into branches until every branch ends in a prime 'leaf'. For example, 60=2×2×3×560 = 2 \times 2 \times 3 \times 5 or 22×3×52^2 \times 3 \times 5.

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Highest Common Factor (HCF): The HCF (also known as GCD) of two or more numbers is the largest number that divides each of them exactly. In a visual Venn diagram representing the prime factors of two numbers, the HCF is the product of the prime factors located in the overlapping central intersection.

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Lowest Common Multiple (LCM): The LCM is the smallest non-zero number that is a multiple of all given numbers. On a number line, if two grasshoppers jump in intervals of 44 units and 66 units respectively, the first point where they both land is 1212, which is their LCM.

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Relationship between HCF and LCM: For any two given natural numbers, the product of the HCF and LCM is always equal to the product of the two numbers. This is a fundamental property used to find a missing value when the other three are known.

📐Formulae

Product of two numbers=HCF×LCM\text{Product of two numbers} = \text{HCF} \times \text{LCM}

LCM=First Number×Second NumberHCF\text{LCM} = \frac{\text{First Number} \times \text{Second Number}}{\text{HCF}}

HCF=First Number×Second NumberLCM\text{HCF} = \frac{\text{First Number} \times \text{Second Number}}{\text{LCM}}

One Number=HCF×LCMOther Number\text{One Number} = \frac{\text{HCF} \times \text{LCM}}{\text{Other Number}}

💡Examples

Problem 1:

Find the HCF and LCM of 2424 and 3636 using the prime factorization method.

Solution:

Step 1: Write the prime factorization of each number. 24=2×2×2×3=23×3124 = 2 \times 2 \times 2 \times 3 = 2^3 \times 3^1 36=2×2×3×3=22×3236 = 2 \times 2 \times 3 \times 3 = 2^2 \times 3^2 Step 2: To find the HCF, take the product of the common prime factors with the lowest power. HCF=22×31=4×3=12\text{HCF} = 2^2 \times 3^1 = 4 \times 3 = 12 Step 3: To find the LCM, take the product of all prime factors with their highest power. LCM=23×32=8×9=72\text{LCM} = 2^3 \times 3^2 = 8 \times 9 = 72

Explanation:

The prime factorization method breaks numbers into their building blocks. HCF focuses on what they share (minimum powers), while LCM ensures all factors are represented (maximum powers).

Problem 2:

The HCF of two numbers is 1616 and their product is 30723072. Find their LCM.

Solution:

Step 1: Use the relationship formula: HCF×LCM=Product of two numbers\text{HCF} \times \text{LCM} = \text{Product of two numbers} Step 2: Substitute the known values into the formula: 16×LCM=307216 \times \text{LCM} = 3072 Step 3: Solve for LCM: LCM=307216\text{LCM} = \frac{3072}{16} LCM=192\text{LCM} = 192

Explanation:

By applying the property that the product of HCF and LCM equals the product of the two numbers, we can find the LCM by dividing the product of the numbers by their HCF.