Correct Answer

⚙️ TE • Technical LCM Divisibility MethodMathematicsArithmetic

Option A

Quick Summary:

Check divisibility by the factors of the numbers: 12 is $3 \times 4$ and 14 is $2 \times 7$ . Thus, the number must be divisible by $3$ , $4$ , and $7$ . 84 is the smallest, hence test it first.

📐MAMath SolutionLCM Divisibility Method

📋 Given

Divisors are 14 and 12. Options are 84, 126, 168, 210.

🔢 Formula Used

LCM(a,b)=a×bHCF(a,b)\text{LCM}(a, b) = \frac{a \times b}{\text{HCF}(a, b)}LCM(a,b)=HCF(a,b)a×b​

⚡ Exam Hall Shortcut / Speed Trick

Check divisibility by the factors of the numbers: 12 is $3 \times 4$ and 14 is $2 \times 7$ . Thus, the number must be divisible by $3$ , $4$ , and $7$ . 84 is the smallest, hence test it first.

⚠️ Common Student Trap / Pitfall

Many students check only one number (e.g., 14) and stop, forgetting that the number must be a common multiple of both 14 and 12.

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Find Prime Factorization

Break down 14 and 12 into their prime factors to find the Least Common Multiple (LCM).

14=2×7,12=22×314 = 2 \times 7, \quad 12 = 2^2 \times 314=2×7,12=22×3

Calculate LCM

The LCM is the product of the highest powers of all prime factors present.

LCM(14,12)=22×3×7=4×3×7=84\text{LCM}(14, 12) = 2^2 \times 3 \times 7 = 4 \times 3 \times 7 = 84LCM(14,12)=22×3×7=4×3×7=84

Verification

A number is divisible by both 14 and 12 if and only if it is a multiple of their LCM, 84.

84÷14=6,84÷12=784 \div 14 = 6, \quad 84 \div 12 = 784÷14=6,84÷12=7

✅

A is correct because 84 is the smallest number (the LCM) that is exactly divisible by both 14 and 12.

Core Concepts Used

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Least Common Multiple Prime Factorization Divisibility Rules

💡 EXAM TIP

This concept is foundational for 'Time and Work' problems where you calculate the LCM of individual work days to determine total units of work.

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