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Finding Common Ground - The Greatest of All

Grade 7CBSE

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

🔑Concepts

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A factor of a number is an exact divisor of that number. Every number is a factor of itself, and 11 is a factor of every number.

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A multiple of a number is obtained by multiplying it by a natural number. Every number is a multiple of itself and 11.

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A Prime Number is a number that has exactly two factors: 11 and the number itself. Examples: 2,3,5,7,11,…2, 3, 5, 7, 11, \dots.

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Prime Factorization is the process of expressing a composite number as a product of its prime factors. Example: 12=2×2×3=22×312 = 2 \times 2 \times 3 = 2^2 \times 3.

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The Highest Common Factor (HCF), also known as Greatest Common Divisor (GCD), of two or more numbers is the largest number that divides each of them exactly.

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The Least Common Multiple (LCM) of two or more numbers is the smallest number which is a multiple of each of the given numbers.

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For any two positive integers aa and bb, the product of the numbers is always equal to the product of their HCF and LCM.

📐Formulae

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

LCM=Product of two numbersHCFLCM = \frac{\text{Product of two numbers}}{HCF}

HCF=Product of two numbersLCMHCF = \frac{\text{Product of two numbers}}{LCM}

Other Number=HCF×LCMGiven Number\text{Other Number} = \frac{HCF \times LCM}{\text{Given Number}}

💡Examples

Problem 1:

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

Solution:

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 The common prime factors are 22 and 33. We take the lowest power of these common factors: HCF=22×31=4×3=12HCF = 2^2 \times 3^1 = 4 \times 3 = 12.

Explanation:

To find the HCF, identify the common prime factors in both numbers and multiply them using their lowest exponents found in the factorizations.

Problem 2:

Find the LCM of 1212 and 1818.

Solution:

12=22×3112 = 2^2 \times 3^1 18=21×3218 = 2^1 \times 3^2 To find the LCM, we take the highest power of all prime factors present in the numbers: LCM=22×32=4×9=36LCM = 2^2 \times 3^2 = 4 \times 9 = 36.

Explanation:

The Least Common Multiple is calculated by taking each prime factor that appears in any of the factorizations at its highest power.

Problem 3:

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

Solution:

Given: HCF=16HCF = 16 Product=3072\text{Product} = 3072 Using the formula: LCM=Product of numbersHCFLCM = \frac{\text{Product of numbers}}{HCF} LCM=307216LCM = \frac{3072}{16} By division: 3072−16001472−144032−320\begin{array}{r} 3072 \\ - 1600 \\ \hline 1472 \\ - 1440 \\ \hline 32 \\ - 32 \\ \hline 0 \end{array} LCM=192LCM = 192.

Explanation:

We use the fundamental relationship between HCF and LCM. Since HCF×LCM=ProductHCF \times LCM = \text{Product}, we divide the product by the HCF to find the LCM.