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Factors and Multiples - Least Common Multiple (LCM)

Grade 5ICSE

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

🔑Concepts

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Multiples of a number are obtained by multiplying the number by 1,2,3,…1, 2, 3, \dots and so on. For example, multiples of 55 are 5,10,15,20,…5, 10, 15, 20, \dots. Visually, you can think of multiples as equal-sized jumps along a number line starting from 00.

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Common Multiples are the numbers that are multiples of two or more given numbers. If you visualize a Venn diagram with two circles representing the sets of multiples for two different numbers, the common multiples would be located in the overlapping central region of the circles.

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The Least Common Multiple (LCM) is the smallest non-zero common multiple of two or more numbers. It represents the very first point where the 'jumps' of different-sized steps on a number line would land on the exact same spot.

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The Listing Method involves writing down the multiples of each number in a sequence until the smallest common value is identified. For instance, to find the LCM of 44 and 66, we list 4:{4,8,12,16}4: \{4, 8, 12, 16\} and 6:{6,12,18}6: \{6, 12, 18\}, identifying 1212 as the first match.

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The Prime Factorization Method involves breaking each number down into its prime factors using a Factor Tree. A Factor Tree looks like a branching structure where a number (the trunk) splits into two factors (branches) until only prime numbers (the leaves) remain. The LCM is the product of the highest powers of all prime factors involved.

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The Common Division Method (Ladder Method) involves writing the numbers in a horizontal row and dividing them by the smallest possible prime number that divides at least one of them. You bring down numbers that are not divisible. This creates a ladder-like visual structure. The LCM is the product of all the divisors used on the left side of the 'ladder'.

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For Co-prime numbers (numbers that have no common factor other than 11, such as 88 and 99), the LCM is always equal to the product of the two numbers. Visually, since their factors don't overlap, you must multiply them together to find their first shared multiple.

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The relationship between HCF and LCM states that for any two numbers, the product of their HCF and LCM is equal to the product of the two numbers themselves. This can be visualized as a balance where HCF×LCMHCF \times LCM on one side equals Number1×Number2Number_1 \times Number_2 on the other.

📐Formulae

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

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

Product of two numbers=a×b\text{Product of two numbers} = a \times b

LCM(Co-prime numbers a,b)=a×bLCM(\text{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 every prime factor present (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 together. LCM=22×32=4×9=36LCM = 2^2 \times 3^2 = 4 \times 9 = 36.

Explanation:

We break the numbers into their prime 'building blocks' and take the maximum frequency of each block to ensure the resulting multiple is divisible by both original numbers.

Problem 2:

The HCF of two numbers is 66 and their product is 432432. Find their LCM.

Solution:

Step 1: Use the relationship formula: LCM×HCF=Product of two numbersLCM \times HCF = \text{Product of two numbers} Step 2: Substitute the known values into the formula: LCM×6=432LCM \times 6 = 432 Step 3: Solve for LCM by dividing the product by the HCF: LCM=4326LCM = \frac{432}{6} LCM=72LCM = 72.

Explanation:

This approach uses the mathematical property that links the highest common factor and the least common multiple to find an unknown value without knowing the specific numbers.