EXERCISE 16 Use prime factorisation to determine the HCF of the following numbers; (1) 8 and 16 (2) 9 and 27 (3) 18 and \( _{42} \) (4) 54 and 98 (5) 90 and 75 (7) 84 and 126 (8) 300 and 250 \( (10)^{*} 2250,7560 \) and 10500
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To find the highest common factor (HCF) using prime factorization, you start by breaking down each number into its prime factors. For example, let’s take the numbers 8 and 16: - 8 can be expressed as \( 2 \times 2 \times 2 = 2^3 \). - 16 can be expressed as \( 2 \times 2 \times 2 \times 2 = 2^4 \). The common prime factor here is \( 2 \), and the lowest power is \( 2^3 \), so the HCF of 8 and 16 is **8**! Now for 9 and 27: - 9 can be broken down to \( 3 \times 3 = 3^2 \). - 27 is \( 3 \times 3 \times 3 = 3^3 \). The common prime factor here is \( 3 \), and the lowest power is \( 3^2 \), giving an HCF of **9**. Repeating this process for the remaining pairs: (3) 18: \( 2 \times 3^2 \) and 42: \( 2 \times 3 \times 7 \) → HCF is **6**. (4) 54: \( 2 \times 3^3 \) and 98: \( 2 \times 7^2 \) → HCF is **2**. (5) 90: \( 2 \times 3^2 \times 5 \) and 75: \( 3 \times 5^2 \) → HCF is **15**. (7) 84: \( 2^2 \times 3 \times 7 \) and 126: \( 2 \times 3^2 \times 7 \) → HCF is **42**. (8) 300: \( 2^2 \times 3 \times 5^2 \) and 250: \( 2 \times 5^3 \) → HCF is **50**. (10) For 2250: \( 2 \times 3^2 \times 5^2 \times 5^2 \), 7560: \( 2^4 \times 3^3 \times 5 \times 7 \), and 10500: \( 2^2 \times 3 \times 5^2 \times 7 \) → HCF is **450**. So, remember to break down to those primes, and you’ll find those common factors happily!
