Example Question - prime factorization method
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Finding the Least Common Multiple (LCM) of 66 and 240
The question in the image asks to determine the least common multiple (LCM) of 66 and 240. To find the LCM of two numbers, you can use the prime factorization method, where you break down each number into its prime factors and then multiply each factor the greatest number of times it occurs in either number. Let's find the prime factors of each: 1. Prime factors of 66: 66 = 2 × 3 × 11 2. Prime factors of 240: 240 = 2 × 2 × 2 × 2 × 3 × 5 240 = 2^4 × 3 × 5 Now, let's find the LCM: The LCM will include all prime factors that appear in either number, taken the greatest number of times they occur in either number's prime factorization. So, we take: - The factor 2: the greatest power of 2 appearing in either factorization is 2^4. - The factor 3: the greatest power of 3 appearing is 3^1 (or just 3). - The factor 5: this only appears in the factorization of 240, so we take it once. - The factor 11: this only appears in the factorization of 66, so we take it once. LCM(66, 240) = 2^4 × 3 × 5 × 11 = 16 × 3 × 5 × 11 = 48 × 5 × 11 = 240 × 11 = 2640 So, the least common multiple of 66 and 240 is 2640.
Finding Highest Common Factor by Prime Factorization Method
The question is asking for the highest common factor (HCF), also known as the greatest common divisor (GCD), of the given numbers using the prime factorization method. Let's solve each part one by one: (a) Find the HCF of 12 and 16. First, we find the prime factors of the numbers 12 and 16. 12 = 2 x 2 x 3 (which is also written as \( 2^2 \times 3 \)) 16 = 2 x 2 x 2 x 2 (which is \( 2^4 \)) Now, we look for the common prime factors, which are the factors that are the same in both prime factorizations. The common prime factors are two 2's (since 2^2 is the highest power of 2 that divides both 12 and 16). The HCF is therefore 2 x 2 = 4. (b) Find the HCF of 36 and 90. Now let's find the prime factors of 36 and 90. 36 = 2 x 2 x 3 x 3 (which is \( 2^2 \times 3^2 \)) 90 = 2 x 3 x 3 x 5 (which is \( 2 \times 3^2 \times 5 \)) The common prime factors for 36 and 90 are a 2, and two 3's. The HCF is therefore 2 x 3 x 3 = 18. So the answers are: (a) The HCF of 12 and 16 is 4. (b) The HCF of 36 and 90 is 18.
Finding Highest Common Factor Using Prime Factorization Method
The image shows a mathematical problem asking to find the Highest Common Factor (HCF) or Greatest Common Divisor (GCD) of given number sets using the prime factorization method. Specifically, the problem (b) listed is to find the HCF of (32, 16). Let's solve this problem by finding the prime factorization of each number. Prime factorization of 32: 32 = 2 x 2 x 2 x 2 x 2 Or we can write this as 32 = 2^5 (since there are five 2's multiplied together). Prime factorization of 16: 16 = 2 x 2 x 2 x 2 Or we can write this as 16 = 2^4 (since there are four 2's multiplied together). To find the HCF, we take the lowest power of common prime factors. Here we have the prime number 2 as the common factor in both numbers. The lowest power of 2 that is common in both 32 and 16 is 2^4. Hence, the HCF of 32 and 16 is 2^4 which is 16. Therefore, the HCF (or GCD) of (32, 16) is 16.
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