HCF Concepts

1. HCF of Numbers

• Idea: Finding the greatest number that divides two or more numbers exactly.

• Formula:

$$\text{HCF} = \text{Greatest common factor}$$

• Example: Find the HCF of 36, 48 and 60.

  Solution:

  36 = 2² × 3²
  48 = 2⁴ × 3
  60 = 2² × 3 × 5

  Common factors:

$$2^2\times3=12$$

Answer: 12

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2. Greatest Measure / Largest Size

• Idea: Finding the largest length, capacity or size that can exactly measure or divide several given quantities.

• Formula:

$$\text{Greatest measure}=\text{HCF of the given quantities}$$

• Example: Find the largest rod that can measure 42 m, 63 m and 84 m exactly.

  Solution:

$$HCF(42,63,84)=21$$

Answer: 21 m

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3. Maximum Equal Groups

• Idea: Dividing different quantities into equal groups of the largest possible size.

• Formula:

$$\text{Maximum group size}=\text{HCF of the quantities}$$

• Example: Three stacks contain 96, 144 and 240 books. Find the largest possible number of books in each equal pile.

  Solution:

$$HCF(96,144,240)=48$$

Answer: 48 books per pile

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4. Greatest Number Leaving the Same Remainder

• Idea: Finding the greatest number that divides several numbers and leaves the same remainder.

• Formula:

$$\text{Required number} ====================== HCF(a-r,b-r,c-r)$$

• Example: Find the greatest number that divides 867 and 255, leaving remainder 3 in each case.

  Solution:

$$867-3=864$$ $$255-3=252$$

Therefore,

$$HCF(864,252)=36$$

Answer: 36

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5. Greatest Number Leaving Different Remainders

• Idea: Finding the greatest number that leaves different specified remainders.

• Formula:

$$\text{Required number} ====================== HCF(a-r_1,b-r_2,c-r_3)$$

• Example: Find the greatest number that divides 65, 89 and 113 leaving remainders 2, 5 and 8 respectively.

  Solution:

$$65-2=63$$ $$89-5=84$$ $$113-8=105$$

Therefore,

$$HCF(63,84,105)=21$$

Answer: 21

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6. HCF of Fractions

• Idea: Finding the HCF when the given numbers are fractions.

• Formula:

$$HCF\left(\frac ab,\frac cd\right) ================================= \frac{HCF(a,c)}{LCM(b,d)}$$

• Example: Find the HCF of

$$\frac{5}{12},\frac{7}{18}$$

Solution:

$$HCF(5,7)=1$$ $$LCM(12,18)=36$$

Therefore,

$$HCF=\frac1{36}$$

Answer: (\frac1{36})

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7. HCF of Decimals

• Idea: Removing decimal places and finding the HCF of the resulting integers.

• Formula:

  Multiply all numbers by the same power of 10 until they become integers.

• Example: Find the HCF of 1.2, 1.8 and 2.4.

  Solution:

  Multiply all by 10:

  1.2 → 12
  1.8 → 18
  2.4 → 24

$$HCF(12,18,24)=6$$

Divide by 10:

$$HCF=0.6$$

Answer: 0.6

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8. HCF Using Euclidean Algorithm

• Idea: Quickly finding the HCF of large numbers using repeated division.

• Formula:

$$HCF(a,b)=HCF(b,a\bmod b)$$

• Example: Find the HCF of 867 and 255.

  Solution:

  867 = 255 × 3 + 102
  255 = 102 × 2 + 51
  102 = 51 × 2 + 0

  Therefore:

$$HCF=51$$

Answer: 51

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Advanced

9. HCF of Algebraic Expressions

• Idea: Finding the common factor of algebraic expressions.

• Formula:

  Factorise each expression and take the common factors with the smallest powers.

• Example: Find the HCF of

$$x^2-9$$

and

$$x^2-6x+9$$

Solution:

$$x^2-9=(x-3)(x+3)$$ $$x^2-6x+9=(x-3)^2$$

Common factor:

$$x-3$$

Answer: (x-3)

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10. HCF–LCM Relationship

• Idea: Finding one of HCF or LCM when the other and the numbers are known.

• Formula:

$$HCF(a,b)\times LCM(a,b)=a\times b$$

• Example: The HCF of two numbers is 12 and their LCM is 180. If one number is 36, find the other.

  Solution:

$$12\times180=36\times x$$ $$x=\frac{2160}{36}=60$$

Answer: 60