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

Option B: $2^3 \times 3^4 \times 5^4$

Option C: $2^2 \times 3^2 \times 5^4$

Option D: $2^3 \times 3^2 \times 5^2$

What is the Greatest Common Divisor (GCD) of 2^3 * 3^2 * 5^4 a... | Mathematics MCQ | Examoogle

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What is the Greatest Common Divisor (GCD) of 2³ * 3² * 5⁴ and 2² * 3⁴ * 5²?

$2^2 \times 3^2 \times 5^2$

$2^3 \times 3^4 \times 5^4$

$2^2 \times 3^2 \times 5^4$

$2^3 \times 3^2 \times 5^2$

Correct Answer

📐 MA • Math Prime FactorizationMathematicsMathematics

Option A

$2^2 \times 3^2 \times 5^2$

Quick Summary:

To find the GCD of numbers in prime factored form, simply select the smallest exponent for each common prime base present in the factors.

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📐MAMath SolutionPrime Factorization

📋 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

GCD(ax⋅by⋅cz,ap⋅bq⋅cr)=amin⁡(x,p)⋅bmin⁡(y,q)⋅cmin⁡(z,r)\text{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)}GCD(ax⋅by⋅cz,ap⋅bq⋅cr)=amin(x,p)⋅bmin(y,q)⋅cmin(z,r)

⚡ Exam Hall Shortcut / Speed Trick

To find the GCD of numbers in prime factored form, simply select the smallest exponent for each common prime base present in the factors.

⚠️ Common Student Trap / Pitfall

Students often confuse GCD with LCM and mistakenly pick the maximum exponents instead of the minimum exponents.

📊 Diagram / Illustration

🔢 Step-by-Step Solution

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 $N_2$ .

N1:23,32,54;N2:22,34,52N_1: 2^3, 3^2, 5^4; \quad N_2: 2^2, 3^4, 5^2N1​:23,32,54;N2​:22,34,52

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 $5$ , $\min(4, 2) = 2$ .

min⁡(3,2)=2,min⁡(2,4)=2,min⁡(4,2)=2\min(3, 2) = 2, \quad \min(2, 4) = 2, \quad \min(4, 2) = 2min(3,2)=2,min(2,4)=2,min(4,2)=2

Calculate GCD

Combine the bases with their respective minimum exponents to get the GCD.

GCD=22imes32imes52\text{GCD} = 2^2 imes 3^2 imes 5^2GCD=22imes32imes52

✅

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

Core Concepts Used

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Fundamental Theorem of Arithmetic Greatest Common Divisor (GCD) Prime Factorization

💡 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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