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{:[P=2^(2)×3^(2)×5],[Q=2^(3)×3×7]:} Find the highest common factor (HCF) of P and 5Q

P=22×32×5Q=23×3×7 \begin{array}{l} P=2^{2} \times 3^{2} \times 5 \\ Q=2^{3} \times 3 \times 7 \end{array} \newlineFind the highest common factor (HCF) of P P and 5Q 5 \boldsymbol{Q}

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Q. P=22×32×5Q=23×3×7 \begin{array}{l} P=2^{2} \times 3^{2} \times 5 \\ Q=2^{3} \times 3 \times 7 \end{array} \newlineFind the highest common factor (HCF) of P P and 5Q 5 \boldsymbol{Q}
  1. Given Numbers: We are given two numbers PP and QQ:P=22×32×5P = 2^2 \times 3^2 \times 5Q=23×3×7Q = 2^3 \times 3 \times 7To find the HCF of PP and 5Q5Q, we first need to express 5Q5Q in its prime factorized form.5Q=5×(23×3×7)5Q = 5 \times (2^3 \times 3 \times 7)Now, express 5Q5Q with its prime factors:5Q=23×3×5×75Q = 2^3 \times 3 \times 5 \times 7
  2. Express 5Q5Q in Prime Factors: The HCF of two numbers is the product of the smallest powers of common prime factors present in both numbers.\newlineFor PP and 5Q5Q, the common prime factors are 22 and 33.\newlineThe smallest power of 22 in PP is 222^2, and in 5Q5Q is 232^3.\newlineThe smallest power of 33 in PP is PP22, and in 5Q5Q is 33.\newlineThere is no need to consider the prime factor PP55 for HCF as it is not common in both PP and 5Q5Q.
  3. Calculate Common Prime Factors: Now, calculate the HCF using the smallest powers of the common prime factors:\newlineHCF(P,5Q)=22×3(P, 5Q) = 2^2 \times 3\newlinePerform the multiplication to find the HCF:\newlineHCF(P,5Q)=4×3(P, 5Q) = 4 \times 3\newlineHCF(P,5Q)=12(P, 5Q) = 12
  4. Calculate HCF: The HCF of PP and 5Q5Q is 1212.

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