Correct Answer

📐 MA • Math Direct FormulaMathematicsNumber System

Option B

Quick Summary:

Given: Two numbers 18 and n have LCM = 72 and HCF = 6.

📐MAMath SolutionDirect Formula

📋 Given

Two numbers 18 and n have LCM = 72 and HCF = 6.

🔢 Formula Used

LCM(a,b)×HCF(a,b)=a×bLCM(a, b) \times HCF(a, b) = a \times bLCM(a,b)×HCF(a,b)=a×b

⚡ Exam Hall Shortcut / Speed Trick

Use the property that the product of the two numbers equals the product of their HCF and LCM. Since $18 \times n = 72 \times 6$ , simply divide $432$ by $18$ to get $24$ .

⚠️ Common Student Trap / Pitfall

Students often mistakenly assume the product of two numbers is just the product of their LCM and HCF without realizing that the given variables $a$ and $b$ must satisfy this specific relationship.

📊 Diagram / Illustration

🔢 Step-by-Step Solution

State the relationship formula

For any two positive integers $a$ and $b$ , the product of their HCF and LCM is equal to the product of the numbers themselves.

LCM(a,b)×HCF(a,b)=a×bLCM(a, b) \times HCF(a, b) = a \times bLCM(a,b)×HCF(a,b)=a×b

Substitute the given values

Given $a = 18$ , $LCM = 72$ , and $HCF = 6$ . Substituting these into the formula, we get $72 \times 6 = 18 \times n$ .

72×6=18×n72 \times 6 = 18 \times n72×6=18×n

Solve for n

Isolate $n$ by dividing both sides by $18$ : $n = \frac{72 \times 6}{18}$ . Simplifying this, $n = 4 \times 6 = 24$ .

n=72×618=24n = \frac{72 \times 6}{18} = 24n=1872×6​=24

✅

B is correct because calculating the product of HCF and LCM ( $72 \times 6 = 432$ ) and dividing by 18 yields $24$ .

Core Concepts Used

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HCF and LCM relationship Number systems Algebraic division

💡 EXAM TIP

This fundamental identity $a \times b = HCF \times LCM$ is frequently used in algebraic simplification and polynomial divisibility problems in higher-level competitive mathematics.

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