What are the highest common factor (HCF) and the least common multiple (LCM) of 120 and 90?

To find the Highest Common Factor (HCF) and the Least Common Multiple (LCM) of the numbers 120 and 90, we can follow a systematic approach.

Finding the HCF

The HCF is the largest number that divides both numbers without leaving a remainder. One method to find the HCF is through prime factorization.

Step 1: Prime Factorization

Start by breaking down each number into its prime factors:

• 120:
  • 120 = 2 × 60
  • 60 = 2 × 30
  • 30 = 2 × 15
  • 15 = 3 × 5

  So, the prime factorization of 120 is: 2³ × 3¹ × 5¹

• 90:
  • 90 = 2 × 45
  • 45 = 3 × 15
  • 15 = 3 × 5

  So, the prime factorization of 90 is: 2¹ × 3² × 5¹

Step 2: Identify Common Factors

Now, we identify the common prime factors and take the lowest exponent for each:

• Common factors are 2, 3, and 5.
• For 2: minimum exponent is 1 (from 90).
• For 3: minimum exponent is 1 (from 120).
• For 5: minimum exponent is 1 (from both).

Step 3: Calculate HCF

Therefore, the HCF is:

HCF = 2¹ × 3¹ × 5¹ = 2 × 3 × 5 = 30

Finding the LCM

The LCM is the smallest number that both numbers can divide without leaving a remainder. Again, we can use the prime factorization method.

Step 1: Use Prime Factors

For the LCM, we take all prime factors with the highest exponent:

• For 2: highest exponent is 3 (from 120).
• For 3: highest exponent is 2 (from 90).
• For 5: highest exponent is 1 (from both).

Step 2: Calculate LCM

Thus, the LCM is:

LCM = 2³ × 3² × 5¹

LCM = 8 × 9 × 5 = 360

Final Results

In conclusion:

• HCF of 120 and 90: 30
• LCM of 120 and 90: 360