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Playing with Numbers - Lowest Common Multiple (LCM)

Grade 6CBSE

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

🔑Concepts

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Multiples: A multiple of a number is the product of that number and any non-zero whole number. For example, multiples of 44 are 4,8,12,16,…4, 8, 12, 16, \dots. Visually, imagine a number line where you start at 00 and take equal jumps of 44 units; every point you land on represents a multiple.

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Common Multiples: When two or more numbers have the same multiple, it is called a common multiple. If you visualize two overlapping circles (a Venn Diagram) where one circle contains multiples of 33 and the other contains multiples of 44, the numbers in the overlapping section (like 12,24,3612, 24, 36) are the common multiples.

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Lowest Common Multiple (LCM): The LCM is the smallest (lowest) number among all the common multiples of two or more given numbers. It is the first number that would appear in the skip-counting lists of all the numbers involved.

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Prime Factorization Method: This method involves expressing each number as a product of its prime factors. The LCM is calculated by multiplying the highest power of every prime factor present in the numbers. Imagine a table where each row lists the prime factors of a number; the LCM captures the maximum number of times each prime appears in any single row.

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Common Division Method: Also known as the 'Ladder Method', this involves writing the numbers in a row and dividing them by the smallest prime number that can divide at least one of them. You continue until the last row consists only of 11s. Visually, this looks like a downward staircase or ladder, and the LCM is the product of all the divisors on the left side.

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LCM of Co-prime Numbers: If two numbers are co-prime (meaning they have no common factors other than 11, like 55 and 77), their LCM is simply their product. For example, LCM(5,7)=5×7=35LCM(5, 7) = 5 \times 7 = 35.

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Relationship between HCF and LCM: For any two numbers, the product of their HCF and LCM is always equal to the product of the two numbers themselves. This can be visualized as a balance scale where a×ba \times b on one side is perfectly balanced by HCF×LCMHCF \times LCM on the other.

📐Formulae

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

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

LCM of co-prime numbers a,b=a×b\text{LCM of co-prime numbers } a, b = a \times b

💡Examples

Problem 1:

Find the LCM of 1212 and 1818 using the Prime Factorization method.

Solution:

Step 1: Write the prime factorization of each number. 12=2×2×3=22×3112 = 2 \times 2 \times 3 = 2^2 \times 3^1 18=2×3×3=21×3218 = 2 \times 3 \times 3 = 2^1 \times 3^2 Step 2: Identify the highest power of each prime factor present in the factorizations. The primes involved are 22 and 33.

  • The highest power of 22 is 222^2.
  • The highest power of 33 is 323^2. Step 3: Multiply these highest powers to find the LCM. LCM=22×32=4×9=36LCM = 2^2 \times 3^2 = 4 \times 9 = 36

Explanation:

This method finds the smallest number that 'contains' all the prime building blocks of both numbers in their required quantities.

Problem 2:

Find the LCM of 20,25, and 3020, 25, \text{ and } 30 using the Common Division method.

Solution:

Step 1: Arrange the numbers in a row and divide by the smallest prime factor 22: 2∣20,25,302 | 20, 25, 30 ∣10,25,15 | 10, 25, 15 Step 2: Divide by 22 again (as 1010 is divisible by 22): 2∣5,25,152 | 5, 25, 15 Step 3: Divide by the next prime factor 33: 3∣5,25,53 | 5, 25, 5 Step 4: Divide by the prime factor 55: 5∣1,5,15 | 1, 5, 1 Step 5: Divide by 55 again: 5∣1,1,15 | 1, 1, 1 Step 6: Multiply all the divisors on the left: LCM=2×2×3×5×5=4×3×25=300LCM = 2 \times 2 \times 3 \times 5 \times 5 = 4 \times 3 \times 25 = 300

Explanation:

The common division method is a systematic way to reduce multiple numbers to 11 simultaneously, ensuring all unique and shared prime factors are accounted for in the final product.