LCM Through Prime Factorization: CBSE Class 10 Sample Paper 2025 Q1

Are you preparing for your CBSE Class 10 Mathematics board exam? If so, you’ve probably noticed that the Real Numbers chapter forms the foundation of many questions in your exam. Today, we’re going to dive deep into Question 1 from the official CBSE Class 10 Standard Level Sample Question Paper 2025 – a beautifully crafted 1-mark question that tests your understanding of LCM (Least Common Multiple) – finding LCM through prime factorization.

While this might seem like a simple question at first glance, it perfectly encapsulates how the CBSE board tests your conceptual understanding rather than just rote memorization. Let’s break down this problem step by step and understand not just how to solve it, but why each step matters.

The Question

Here’s what you’re asked to solve:

If a = 2² × 3^x, b = 2² × 3 × 5, and c = 2² × 3 × 7, and the LCM(a, b, c) = 3780, find the value of x.

1. 1
2. 2
3. 3
4. 0

At first, this might look intimidating with all those variables and exponents. But trust me, once you understand the systematic approach, questions like these become incredibly straightforward!

This question tests your ability to:

1. Apply the prime factorization method correctly
2. Understand the concept of LCM as “highest powers of all prime factors”
3. Work backwards from a given result to find unknown values
4. Think logically and systematically

Key Concepts: A Quick Refresher

What is Prime Factorization?

Prime factorization is the process of breaking down a number into its basic building blocks – prime numbers. For example, 12 = 2² × 3. Every composite number can be uniquely expressed as a product of prime numbers.

What is LCM?

The Least Common Multiple (LCM) of two or more numbers is the smallest number that is divisible by all of them. When finding LCM using prime factorization, we take the highest power of each prime factor present in any of the numbers.

This is the golden rule that makes this question solvable!

The Three-Step Solution

Let’s solve this problem using a systematic three-step approach that will work for any similar question.

Step 1: Find the LCM of a, b, and c Using Prime Factorization

First, let’s list out what we have:

• a = 2² × 3^x
• b = 2² × 3¹ × 5¹
• c = 2² × 3¹ × 7¹

Now, to find the LCM, we need to identify all the prime factors present across all three numbers:

• Prime factors found: 2, 3, 5, 7

Next, we apply the LCM rule – take the highest power of each prime:

For prime 2: The highest power is 2² (same in all three numbers)

For prime 3: We have 3^x in ‘a’ and 3¹ in both ‘b’ and ‘c’. Now here’s the tricky part – we don’t know what x is yet! If x is 0 or 1, then the highest power would be 1. But if x is greater than 1, then the highest power would be x. So we’ll leave this as a question mark for now.

For prime 5: It appears only in ‘b’ with power 1, so we take 5¹

For prime 7: It appears only in ‘c’ with power 1, so we take 7¹

Therefore, LCM(a, b, c) = 2² × 3^? × 5¹ × 7¹

Step 2: Prime Factorize 3780

We’re told that this LCM equals 3780. So let’s break down 3780 into its prime factors using the systematic division method:

3780 ÷ 2 = 1890;
1890 ÷ 2 = 945;
945 ÷ 3 = 315;
315 ÷ 3 = 105;
105 ÷ 3 = 35;
35 ÷ 5 = 7;
7 ÷ 7 = 1

Counting the prime factors: 3780 = 2² × 3³ × 5¹ × 7¹

Step 3: Equate and Solve for x

Now comes the beautiful part – we have two expressions for the same LCM:

From Step 1: LCM = 2² × 3^? × 5¹ × 7¹
From Step 2: 3780 = 2² × 3³ × 5¹ × 7¹

Since both expressions represent the same number, we can equate the powers of each prime factor:

• Powers of 2: 2 = 2 ✓
• Powers of 3: ? = 3
• Powers of 5: 1 = 1 ✓
• Powers of 7: 1 = 1 ✓

Therefore, x = 3

And since x = 3 is greater than 1, our earlier assumption about the highest power of 3 being ‘x’ was correct!

Answer: x = 3 (Option C)

Common Mistakes to Avoid

1. Forgetting the LCM Rule: Some students mistakenly multiply all the powers instead of taking the highest power of each prime. Remember: LCM uses the HIGHEST power, HCF uses the LOWEST power.
2. Calculation Errors in Prime Factorization: When dividing 3780, take your time. One small calculation error will throw off your entire answer.
3. Not Considering All Cases for x: Some students immediately assume 3^x is the highest power without reasoning through whether x could be less than 1.
4. Rushing Through 1-Mark Questions: Just because it’s worth 1 mark doesn’t mean you should rush. These questions test fundamental concepts that appear in harder problems too. And remember every mark counts toward making the final score a 100.

Pro Tips for CBSE Board Exams

1. Show Your Work: Even for 1-mark questions, write the key steps. Partial marking might save you in case of a silly error.
2. Practice Prime Factorization Speed: Being quick at prime factorization helps in both MCQs and longer questions. Practice factorizing numbers daily.
3. Verify Your Answer: After finding x = 3, quickly substitute it back: LCM(2² × 3³, 2² × 3 × 5, 2² × 3 × 7) should give you 2² × 3³ × 5 × 7 = 3780. This takes 10 seconds and confirms your answer.
4. Understand, Don’t Memorize: Notice how we used logical reasoning throughout? This approach works for any LCM problem, not just this specific question.

Conclusion

Question 1 from the CBSE Class 10 Standard Sample Paper might be a 1-mark question, but it’s packed with important concepts. It tests your understanding of prime factorization, LCM, and algebraic thinking – all crucial skills for your board exam and beyond.

The key takeaway? Systematic approach beats random attempts every time. By following our three-step method – (1) Find LCM with unknown, (2) Prime factorize the given value, (3) Equate and solve – you can tackle any similar question with confidence.

Ready to practice more? Check out our complete video solution series for the CBSE Class 10 Sample Papers, where we break down every single question with the same level of detail and clarity.

Happy learning, and good luck with your board exams! 🌟

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Why This Question Matters

Before we jump into the solution, let’s understand why CBSE includes questions like this in the sample paper. Prime factorization and LCM aren’t just abstract mathematical concepts – they have real-world applications in:

• Computer Science: Algorithm optimization and data structure design
• Cryptography: Security systems rely heavily on prime numbers
• Music Theory: Understanding rhythms and beat patterns
• Scheduling Problems: Finding common time intervals

Keywords: CBSE Class 10 Maths, LCM, Prime Factorization, Real Numbers, Sample Paper 2025, Board Exam Preparation, Standard Mathematics