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Playing with Numbers - Highest Common Factor (HCF)

Grade 6CBSE

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

🔑Concepts

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Factors: A factor of a number is an exact divisor of that number, meaning it divides the number without leaving any remainder. You can visualize factors by imagining a fixed number of dots; for example, the number 1212 can be arranged in a perfect rectangular grid of 2times62 \\times 6, 3times43 \\times 4, or 1times121 \\times 12. Each of these dimensions (1,2,3,4,6,121, 2, 3, 4, 6, 12) represents a factor.

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Common Factors: When we compare the factors of two or more different numbers, the factors that are present in both lists are called common factors. Visually, if you draw a Venn diagram with two overlapping circles representing the sets of factors for two numbers, the common factors are the ones placed in the overlapping middle section.

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Highest Common Factor (HCF): The HCF of two or more numbers is the greatest or largest number among all their common factors. It is also referred to as the Greatest Common Divisor (GCD). In a Venn diagram of factors, the HCF is the largest value found in the intersection of the sets.

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Prime Factorization Method: This method involves breaking down each number into a product of prime numbers. You can visualize this using a 'Factor Tree' where the main number is the root, and it splits into branches of factors until every branch ends in a prime number (the leaves) that cannot be split further.

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Finding HCF via Prime Factorization: Once the prime factorization is complete for all numbers, you identify the prime factors that are common to all of them. The HCF is calculated by multiplying these common prime factors together. For example, if two numbers both have 22 and 33 in their prime factor trees, their HCF must include 2times32 \\times 3.

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Division Method: This is a systematic process for finding the HCF of large numbers. You divide the larger number by the smaller one, then take the remainder and make it the new divisor, while the previous divisor becomes the new dividend. This 'staircase' of divisions continues until the remainder is zero. The last divisor used is the HCF.

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HCF of Co-prime Numbers: Two numbers are called co-prime if their only common factor is 11. For instance, 88 and 1515 are co-prime because no number other than 11 divides both. Consequently, the HCF of any set of co-prime numbers is always 11.

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Properties of HCF: The HCF of a group of numbers is never greater than the smallest number in that group. It is always a factor of each of the numbers. Visually, the HCF represents the largest possible 'unit' or 'block' that can be used to perfectly measure or build each of the original numbers.

📐Formulae

textHCF=textProductofthelowestpowersofcommonprimefactors\\text{HCF} = \\text{Product of the lowest powers of common prime factors}

textHCF(a,b)timestextLCM(a,b)=atimesb\\text{HCF}(a, b) \\times \\text{LCM}(a, b) = a \\times b

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

💡Examples

Problem 1:

Find the HCF of 2424, 3636, and 4848 using the prime factorization method.

Solution:

Step 1: Write the prime factorization of each number:

  • 24=2times2times2times3=23times3124 = 2 \\times 2 \\times 2 \\times 3 = 2^3 \\times 3^1
  • 36=2times2times3times3=22times3236 = 2 \\times 2 \\times 3 \\times 3 = 2^2 \\times 3^2
  • 48=2times2times2times2times3=24times3148 = 2 \\times 2 \\times 2 \\times 2 \\times 3 = 2^4 \\times 3^1

Step 2: Identify the common prime factors. The common prime factors are 22 and 33.

Step 3: Select the lowest power of each common factor:

  • For 22, the lowest power is 222^2.
  • For 33, the lowest power is 313^1.

Step 4: Multiply these to get the HCF:

  • HCF=22times31=4times3=12HCF = 2^2 \\times 3^1 = 4 \\times 3 = 12.

Explanation:

We break down each number into its prime building blocks and then find the largest combination of blocks that is present in every single number.

Problem 2:

Find the HCF of 144144 and 198198 using the Division Method.

Solution:

Step 1: Divide the larger number (198198) by the smaller number (144144):

  • 198=144times1+54198 = 144 \\times 1 + 54 (Remainder is 5454)

Step 2: Divide the previous divisor (144144) by the remainder (5454):

  • 144=54times2+36144 = 54 \\times 2 + 36 (Remainder is 3636)

Step 3: Divide the previous divisor (5454) by the remainder (3636):

  • 54=36times1+1854 = 36 \\times 1 + 18 (Remainder is 1818)

Step 4: Divide the previous divisor (3636) by the remainder (1818):

  • 36=18times2+036 = 18 \\times 2 + 0 (Remainder is 00)

Step 5: Since the remainder is now 00, the last divisor used is the HCF.

  • HCF=18HCF = 18.

Explanation:

The division method, or Euclidean algorithm, uses the property that the HCF of two numbers also divides their difference. We repeat the division until we find the largest value that leaves no remainder.