Pregunta
Use prime factors to determine the HCF of \( 28 ; 196 \) and 280
Ask by Pearson Olson. in South Africa
Feb 27,2025
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The HCF of 28, 196, and 280 is 28.
Solución
To determine the HCF (Highest Common Factor) using prime factors, follow these steps:
1. Find the prime factors of each number:
• 28 = 2² × 7
• 196 = 2² × 7² (since 196 = (2×7)²)
• 280 = 2³ × 5 × 7
2. Identify the common prime factors from all three numbers:
• For the prime number 2, the lowest exponent is 2 (as both 28 and 196 have 2², while 280 has 2³).
• For the prime number 7, the lowest exponent is 1 (28 has 7¹, 196 has 7², and 280 has 7¹).
• The prime number 5 is not common to all three numbers.
3. Multiply these common prime factors with their lowest exponents:
HCF = 2² × 7¹ = 4 × 7 = 28
Thus, the HCF of 28, 196, and 280 is 28.
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To find the highest common factor (HCF) using prime factors, we start by finding the prime factorization of each number. - For \( 28 \): - \( 28 = 2 \times 14 = 2 \times 2 \times 7 = 2^2 \times 7^1 \) - For \( 196 \): - \( 196 = 2 \times 98 = 2 \times 2 \times 49 = 2^2 \times 7^2 \) - For \( 280 \): - \( 280 = 2 \times 140 = 2 \times 2 \times 70 = 2^3 \times 5 \times 7^1 \) Now, we take the lowest power of all common prime factors: - For \( 2 \): minimum power is \( 2^2 \) - For \( 7 \): minimum power is \( 7^1 \) Thus, the HCF is: \[ HCF = 2^2 \times 7^1 = 4 \times 7 = 28 \] So, the HCF of \( 28 \), \( 196 \), and \( 280 \) is \( 28 \).
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