Find the HCF of 48, 72 and 144. | Mathematics MCQ | Examoogle

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Find the HCF of 48, 72 and 144.

Correct Answer

📐 MA • Math Prime FactorizationMathematicsNumber System

Option B

Quick Summary:

Check the smallest number (48). Since 48 divides 144 but not 72, check the next highest factor of 48, which is 24. Since 24 divides 48, 72, and 144, it is the HCF.

📐MAMath SolutionPrime Factorization

📋 Given

The numbers given are 48, 72, and 144.

🔢 Formula Used

HCF(a,b,c)=Product of the lowest power of each common prime factor\text{HCF}(a, b, c) = \text{Product of the lowest power of each common prime factor}HCF(a,b,c)=Product of the lowest power of each common prime factor

⚡ Exam Hall Shortcut / Speed Trick

Check the smallest number (48). Since 48 divides 144 but not 72, check the next highest factor of 48, which is 24. Since 24 divides 48, 72, and 144, it is the HCF.

⚠️ Common Student Trap / Pitfall

Students often stop at a common factor like 12, failing to realize that 24 is a larger common divisor that divides all three numbers.

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Prime Factorization

Express each number as a product of its prime factors: $48 = 2^4 \times 3^1$ , $72 = 2^3 \times 3^2$ , and $144 = 2^4 \times 3^2$ .

48=24×31,72=23×32,144=24×3248 = 2^4 \times 3^1, \quad 72 = 2^3 \times 3^2, \quad 144 = 2^4 \times 3^248=24×31,72=23×32,144=24×32

Identify Common Factors

Identify the common prime factors (2 and 3) and take the lowest exponent for each from the three expressions.

Lowest power of 2=23,Lowest power of 3=31\text{Lowest power of } 2 = 2^3, \quad \text{Lowest power of } 3 = 3^1Lowest power of 2=23,Lowest power of 3=31

Calculate HCF

Multiply these lowest powers to find the Highest Common Factor.

HCF=23×31=8×3=24\text{HCF} = 2^3 \times 3^1 = 8 \times 3 = 24HCF=23×31=8×3=24

✅

B is correct because the greatest integer that divides 48, 72, and 144 without leaving a remainder is 24.

Core Concepts Used

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Prime Factorization Greatest Common Divisor Divisibility Rules

💡 EXAM TIP

The HCF is always a factor of the difference between any two of the numbers; calculating the difference first can significantly speed up finding the HCF in larger sets.

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