Find the HCF of these pairs of numbers

Find the HCF of these pairs of numbers.

a 180, 300

b 270, 378

The correct answer and explanation is:

To find the Highest Common Factor (HCF) of the given pairs of numbers, we will first list their factors and then identify the greatest number that appears in both sets.

a) 180 and 300

1. Prime factorization of 180:
   • 180 ÷ 2 = 90
   • 90 ÷ 2 = 45
   • 45 ÷ 3 = 15
   • 15 ÷ 3 = 5
   • 5 ÷ 5 = 1
   • Therefore, the prime factorization of 180 is 22×32×52^2 \times 3^2 \times 5.
2. Prime factorization of 300:
   • 300 ÷ 2 = 150
   • 150 ÷ 2 = 75
   • 75 ÷ 3 = 25
   • 25 ÷ 5 = 5
   • 5 ÷ 5 = 1
   • Therefore, the prime factorization of 300 is 22×3×522^2 \times 3 \times 5^2.
3. Finding the HCF:
   The HCF is found by taking the lowest power of each common prime factor:
   • For 2: the lowest power is 222^2.
   • For 3: the lowest power is 313^1.
   • For 5: the lowest power is 515^1.
   Therefore, the HCF of 180 and 300 is 22×31×51=4×3×5=602^2 \times 3^1 \times 5^1 = 4 \times 3 \times 5 = 60.

b) 270 and 378

1. Prime factorization of 270:
   • 270 ÷ 2 = 135
   • 135 ÷ 3 = 45
   • 45 ÷ 3 = 15
   • 15 ÷ 3 = 5
   • 5 ÷ 5 = 1
   • Therefore, the prime factorization of 270 is 2×33×52 \times 3^3 \times 5.
2. Prime factorization of 378:
   • 378 ÷ 2 = 189
   • 189 ÷ 3 = 63
   • 63 ÷ 3 = 21
   • 21 ÷ 3 = 7
   • 7 ÷ 7 = 1
   • Therefore, the prime factorization of 378 is 2×33×72 \times 3^3 \times 7.
3. Finding the HCF:
   The HCF is found by taking the lowest power of each common prime factor:
   • For 2: the lowest power is 212^1.
   • For 3: the lowest power is 333^3.
   Therefore, the HCF of 270 and 378 is 21×33=2×27=542^1 \times 3^3 = 2 \times 27 = 54.

Final Answers:

• The HCF of 180 and 300 is 60.
• The HCF of 270 and 378 is 54.