How to Find HCF and LCM

• The Highest Common Factor (HCF) is the largest factor that two or more numbers share.
• The Lowest Common Multiple (LCM) is the smallest multiple that two or more numbers share.

These concepts are essential for simplifying fractions, solving equations, and much more!

Find the HCF of 12 and 18:

• List the factors of each number.
  • Factors of 12: 1, 2, 3, 4, 6, 12
  • Factors of 18: 1, 2, 3, 6, 9, 18
• Identify the highest common factor → HCF = 6 ✅

Find the LCM of 12 and 18:

• List the multiples of each number.
  • Multiples of 12: 12, 24, 36, 48, …
  • Multiples of 18: 18, 36, 54, 72, …
• Identify the smallest common multiple → LCM = 36 ✅

🔎 Find the HCF and LCM of 180 and 168:

• Step 1: Factorise both numbers into their prime factors:
  $$
  \begin{aligned}
  180 &= 2^2 \times 3^2 \times 5 \\[1em]
  168 &= 2^3 \times 3 \times 7
  \end{aligned}
  $$
• Step 2: Find the HCF (Highest Common Factor) by taking the common factors and using their lowest powers:
  $$ HCF = 2^2 \times 3^1 = 12 $$
• Step 3: Find the LCM (Lowest Common Multiple) by taking all factors and using their highest powers:
  $$ LCM = 2^3 \times 3^2 \times 5 \times 7 = 2520 $$

This method is especially useful when both numbers are large.

🔎 Find the HCF and LCM of 56 and 100:

• Step 1: Factorize both numbers into their prime factors:
  $$
  \begin{aligned}
  56 &= 2^3 \times 7 \\[1em]
  100 &= 2^2 \times 5^2
  \end{aligned}
  $$
• Step 2: Find the HCF (Highest Common Factor) by taking the common factors and using their lowest powers:
  $$ HCF = 2^2 = 4 $$
• Step 3: Find the LCM (Lowest Common Multiple) by taking all factors and using their highest powers:
  $$ LCM = 2^3 \times 5^2 \times 7 = 1400 $$