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Finding Common Ground - Least, but not Last!

Grade 7CBSE

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

🔑Concepts

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

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Prime Factorization Method: To find the LCM, express each number as a product of prime factors. The LCM is the product of the highest powers of all prime factors that appear in any of the numbers.

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Common Division Method: Write the numbers in a row and divide by the smallest prime number that divides at least two of the given numbers. Carry forward numbers that are not divisible. Continue until no two numbers have a common factor other than 11.

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Relationship between HCF and LCM: For any two positive integers aa and bb, the product of the numbers is equal to the product of their Highest Common Factor (HCF) and their Least Common Multiple (LCM).

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If two numbers are co-prime (i.e., their HCF is 11), then their LCM is simply the product of the two numbers.

📐Formulae

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

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

LCM of Fractions=LCM of NumeratorsHCF of Denominators\text{LCM of Fractions} = \frac{\text{LCM of Numerators}}{\text{HCF of Denominators}}

💡Examples

Problem 1:

Find the LCM 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 LCM = 23×32=8×9=722^3 \times 3^2 = 8 \times 9 = 72.

Explanation:

We identify the prime factors involved (22 and 33) and take the highest power of each. The highest power of 22 is 232^3 and the highest power of 33 is 323^2.

Problem 2:

Find the LCM of 12,15,12, 15, and 1818 using the common division method.

Solution:

212,15,1826,15,933,15,931,5,351,5,11,1,1\begin{array}{r|l} 2 & 12, 15, 18 \\ \hline 2 & 6, 15, 9 \\ \hline 3 & 3, 15, 9 \\ \hline 3 & 1, 5, 3 \\ \hline 5 & 1, 5, 1 \\ \hline & 1, 1, 1 \end{array} LCM =2×2×3×3×5=180= 2 \times 2 \times 3 \times 3 \times 5 = 180.

Explanation:

We divide the numbers by prime factors 2,3,2, 3, and 55 sequentially. If a number is not divisible, it is brought down to the next row. The LCM is the product of all divisors.

Problem 3:

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

Solution:

Using the formula: LCM=Product of numbersHCF\text{LCM} = \frac{\text{Product of numbers}}{\text{HCF}} LCM=307216\text{LCM} = \frac{3072}{16} LCM=192\text{LCM} = 192

Explanation:

Since we know the product of two numbers is equal to the product of their HCF and LCM, we divide the given product by the HCF to find the LCM.