What is the Greatest Common Divisor (GCD) of 2^3 * 3^2 * 5^4 and 2^2 * 3^4 * 5^2?

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Examoogle · Accepted Answer

Option A: $2^2 	imes 3^2 	imes 5^2$

Explanation & Concepts:
Given: Two numbers are given as prime factorizations: $N_1 = 2^3 	imes 3^2 	imes 5^4$ and $N_2 = 2^2 	imes 3^4 	imes 5^2$.

Formula Used: $$	ext{GCD}(a^x \cdot b^y \cdot c^z, a^p \cdot b^q \cdot c^r) = a^{\min(x,p)} \cdot b^{\min(y,q)} \cdot c^{\min(z,r)}$$

Steps: Identify prime bases and exponents: The prime bases present in both numbers are $2$, $3$, and $5$. We list their exponents for $N_1$ and $N_2$.. Find the minimum exponent for each base: Comparing exponents: for base $2$, $\min(3, 2) = 2$. For base $3$, $\min(2, 4) = 2$. For base $5$, $\min(4, 2) = 2$.. Calculate GCD: Combine the bases with their respective minimum exponents to get the GCD.

Answer: A is correct because the GCD is obtained by taking the product of the lowest powers of each common prime factor, which results in $2^2 	imes 3^2 	imes 5^2$.

Exam Tip: This method is directly applicable to finding the HCF/GCD of algebraic expressions containing variables, e.g., finding the GCD of $x^2y^3$ and $x^3y^2$.

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Option B: $2^3 	imes 3^4 	imes 5^4$

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Option C: $2^2 	imes 3^2 	imes 5^4$

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Option D: $2^3 	imes 3^2 	imes 5^2$

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