Find the HCF of 2^3 * 3^2 * 5 and 2^2 * 3^3 * 5^2. | Mathematics MCQ | Examoogle

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Find the HCF of 2³ * 3² * 5 and 2² * 3³ * 5².

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

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

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

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

Correct Answer

📐 MA • Math Prime Factorization MethodMathematicsMathematics

Option A

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

Quick Summary:

To find HCF of numbers in prime factored form, simply pick the minimum exponent for every prime base present in both numbers.

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📐MAMath SolutionPrime Factorization Method

📋 Given

Number 1 = 2³ * 3² * 5, Number 2 = 2² * 3³ * 5²

🔢 Formula Used

HCF=Product of the lowest power of each common prime factorHCF = \text{Product of the lowest power of each common prime factor}HCF=Product of the lowest power of each common prime factor

⚡ Exam Hall Shortcut / Speed Trick

To find HCF of numbers in prime factored form, simply pick the minimum exponent for every prime base present in both numbers.

⚠️ Common Student Trap / Pitfall

Confusing HCF with LCM; for LCM, one must take the highest exponent of every prime factor present, leading to the selection of larger powers.

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Identify common prime factors

The prime factors present in both numbers are $2$ , $3$ , and $5$ .

2,3,52, 3, 52,3,5

Compare exponents for base 2

The exponents of $2$ are $3$ and $2$ . The smaller exponent is $2$ .

2min⁡(3,2)=222^{\min(3, 2)} = 2^22min(3,2)=22

Compare exponents for base 3

The exponents of $3$ are $2$ and $3$ . The smaller exponent is $2$ .

3min⁡(2,3)=323^{\min(2, 3)} = 3^23min(2,3)=32

Compare exponents for base 5

The exponents of $5$ are $1$ (implied in $5^1$ ) and $2$ . The smaller exponent is $1$ .

5min⁡(1,2)=515^{\min(1, 2)} = 5^15min(1,2)=51

Calculate the HCF

Multiply the factors with their lowest exponents: $2^2 imes 3^2 imes 5^1$ .

22imes32imes52^2 imes 3^2 imes 522imes32imes5

✅

A is correct because the product of the lowest powers of common prime factors $2$ , $3$ , and $5$ results in $2^2 imes 3^2 imes 5$ .

Core Concepts Used

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Prime Factorization HCF (Highest Common Factor) Exponent Laws

💡 EXAM TIP

This concept is foundational for solving problems involving LCM/HCF of fractions and polynomials in Algebra.

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