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Finding Common Ground - Patterns, Properties, and a Pretty Procedure!

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. For example, the factors of 1818 are 1,2,3,6,9,1, 2, 3, 6, 9, and 1818.

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A Multiple of a number is a number obtained by multiplying it by any natural number. For example, multiples of 55 are 5,10,15,20,…5, 10, 15, 20, \dots

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The Highest Common Factor (HCF) of two or more numbers is the largest number that divides each of them exactly.

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

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

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

📐Formulae

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

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

HCF(a,b)=a×bLCM(a,b)HCF(a, b) = \frac{a \times b}{LCM(a, b)}

💡Examples

Problem 1:

Find the HCF of 2424 and 3636 using the Prime Factorization method.

Solution:

Prime factorization of 2424: 2×2×2×3=23×32 \times 2 \times 2 \times 3 = 2^3 \times 3 Prime factorization of 3636: 2×2×3×3=22×322 \times 2 \times 3 \times 3 = 2^2 \times 3^2 Common prime factors with the lowest powers: 222^2 and 313^1. HCF=22×3=4×3=12HCF = 2^2 \times 3 = 4 \times 3 = 12.

Explanation:

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

Problem 2:

The HCF of two numbers is 66 and their LCM is 7272. If one of the numbers is 1818, find the other number.

Solution:

Let the numbers be a=18a = 18 and bb. We know: HCF×LCM=a×bHCF \times LCM = a \times b 6×72=18×b6 \times 72 = 18 \times b 432=18b432 = 18b b=43218=24b = \frac{432}{18} = 24

Explanation:

We use the property that the product of HCF and LCM of two numbers is equal to the product of the numbers themselves.

Problem 3:

Calculate the difference between the LCM and HCF of 1515 and 2020 using vertical subtraction for the final step.

Solution:

Factors of 15=3×515 = 3 \times 5 Factors of 20=22×520 = 2^2 \times 5 HCF(15,20)=5HCF(15, 20) = 5 LCM(15,20)=22×3×5=60LCM(15, 20) = 2^2 \times 3 \times 5 = 60 Difference =60−5= 60 - 5 60−555\begin{array}{r} 60 \\ -5 \\ \hline 55 \end{array}

Explanation:

First, we find the HCF and LCM using prime factorization. Then, we perform vertical subtraction to find the difference.