Real Numbers is one of the most important chapters in Class 10 Mathematics because it builds the foundation for logical problem-solving, factor-based reasoning, and board-level application questions. Even though the chapter may look simple at first, students often lose marks in previous year questions when they fail to identify whether the question is asking for HCF, LCM, rationalisation logic, or a relationship between numbers.
This chapter is especially scoring when students clearly understand the difference between factors and multiples, the correct use of HCF and LCM, and how to handle rational and irrational numbers in application-based questions. Many board questions are not lengthy, but they are concept-based. That is why strong understanding matters more than memorisation.
At Deeksha Vedantu, we encourage students to approach Real Numbers with clarity and pattern recognition. Once you understand what the question is really asking, most of the numericals become direct and manageable.
Why Real Numbers Is an Important Chapter in Class 10 Maths
Real Numbers is a high-value chapter for Class 10 students because it introduces concepts that are repeatedly used in algebra, number theory, and board exam reasoning.
Key Reasons This Chapter Matters
- It strengthens number sense and logical thinking.
- It includes frequently repeated board-style questions.
- It helps students understand HCF and LCM application clearly.
- It improves accuracy in assertion-reason and case-based questions.
- It forms a conceptual base for higher mathematics.
Topics Commonly Asked from Real Numbers
Students can expect questions from the following areas:
HCF and LCM Based Questions
These are the most common questions in exams. They may be direct, ratio-based, or remainder-based.
Rational and Irrational Numbers
Students may be asked to identify rational and irrational numbers or find a number that makes an expression rational.
Previous Year Application Questions
Board exams often ask practical problems involving teachers, rooms, groups, divisibility, and common arrangements.
Relationship Between HCF and LCM
This is a very important concept. Many direct and indirect questions come from it.
Core Concepts You Must Know Before Solving Questions
Before solving important questions from Real Numbers, students should be clear with the following concepts.
What Is HCF
HCF means Highest Common Factor. It is the greatest number that divides two or more numbers exactly.
What Is LCM
LCM means Lowest Common Multiple. It is the smallest number that is exactly divisible by two or more given numbers.
Difference Between HCF and LCM
HCF is related to factors and usually comes smaller than or equal to the given numbers.
LCM is related to multiples and usually comes greater than or equal to the given numbers.
Relationship Between HCF and LCM
For any two natural numbers:
HCF × LCM = Product of the two numbers
This relation is extremely important for board exam questions.
How to Identify Whether a Question Needs HCF or LCM
Many students know how to calculate HCF and LCM, but they get confused about when to use which one. This is where most mistakes happen.
Use HCF When
Use HCF when the question asks for:
- the greatest number that divides given numbers
- equal grouping
- maximum size possible
- the highest common factor
- division with exact factor logic
Use LCM When
Use LCM when the question asks for:
- the least number divisible by given numbers
- a number that is a common multiple
- repetition or alignment of cycles
- the smallest common arrangement
- remainder-based questions where the unknown number is greater than the given divisors
Important Question 1: Greatest Number Dividing 70 and 125 Leaving Remainders 5 and 8
Find the greatest number which divides 70 and 125 leaving remainders 5 and 8 respectively.
Step 1: Adjust the Numbers
If a number divides 70 leaving remainder 5, then it divides 70 – 5 exactly.
So:
- 70 – 5 = 65
- 125 – 8 = 117
Step 2: Find HCF of 65 and 117
Prime factorisation:
- 65 = 5 × 13
- 117 = 3 × 3 × 13
Common factor = 13
Answer
The required greatest number is 13.
Why This Method Works
This is an HCF question because the same greatest number is dividing both values exactly after removing the remainders.
Important Question 2: Assertion and Reason on Prime Numbers
Assertion: For any two prime numbers p and q, their HCF is 1 and LCM is p + q.
Reason: For any two natural numbers, HCF × LCM is equal to the product of the numbers.
Understanding the Assertion
If two numbers are prime, their HCF is 1. That part is correct.
But the LCM of two prime numbers is not p + q. It is p × q.
So the assertion is false.
Understanding the Reason
The reason is true because for any two natural numbers:
HCF × LCM = Product of the numbers
Answer
Assertion is false, but Reason is true.
Important Question 3: Two Numbers Are in the Ratio 4:5 and Their HCF Is 11
Find the LCM of the two numbers.
Step 1: Form the Numbers
If the numbers are in the ratio 4:5 and their HCF is 11, then the numbers are:
- 11 × 4 = 44
- 11 × 5 = 55
So the numbers are 44 and 55.
Step 2: Use the HCF-LCM Relationship
HCF × LCM = Product of the numbers
11 × LCM = 44 × 55
LCM = 44 × 55 ÷ 11
LCM = 220
Answer
The LCM of the numbers is 220.
Important Question 4: Smallest Irrational Number to Multiply with Root 20 to Get a Rational Number
Find the smallest irrational number by which root 20 should be multiplied so as to get a rational number.
Step 1: Simplify Root 20
root 20 = root (4 × 5)
root 20 = 2 root 5
Step 2: Make the Expression Rational
To remove the irrational part root 5, multiply by root 5.
2 root 5 × root 5 = 2 × 5 = 10
10 is a rational number.
Answer
The required irrational number is root 5.
Important Question 5: Pair of Irrational Numbers Whose Product Is Rational
Choose the pair of irrational numbers whose product is a rational number.
- root 16 and root 4
- root 5 and root 2
- root 3 and root 27
- root 36 and root 2
Check the Correct Option
root 3 is irrational.
root 27 = 3 root 3, which is also irrational.
Now multiply:
root 3 × root 27 = root 81 = 9
9 is a rational number.
Answer
The correct pair is root 3 and root 27.
Important Question 6: LCM of Algebraic Forms with Prime Variables
If two positive integers p and q are expressed in prime factorised form, find their LCM.
Rule for LCM in Prime Factorised Form
Take each prime factor with the highest power appearing in either number.
Example Pattern
If one number contains:
- 2 raised to some power
- 3 raised to some power
- a raised to some power
- b raised to some power
and the other number contains different powers of the same factors, then for LCM:
- choose the greater power of 2
- choose the greater power of 3
- choose the greater power of a
- choose the greater power of b
Key Exam Tip
For LCM, always take the highest powers.
For HCF, always take the common factors with the lowest powers.
Important Question 7: Minimum Number of Rooms for Teachers of Different Subjects
In a teachers’ workshop, the number of teachers teaching French, Hindi, and English are 48, 80, and 144 respectively. Find the minimum number of rooms required if in each room the same number of teachers are seated and all of them are of the same subject.
Step 1: Find the HCF of 48, 80, and 144
Prime factorisation:
- 48 = 2 to the power 4 × 3
- 80 = 2 to the power 4 × 5
- 144 = 2 to the power 4 × 3 to the power 2
Common factor with lowest power = 2 to the power 4 = 16
So, each room will have 16 teachers.
Step 2: Find Total Number of Teachers
48 + 80 + 144 = 272
Step 3: Divide by Number of Teachers per Room
272 ÷ 16 = 17
Answer
The minimum number of rooms required is 17.
Important Question 8: If HCF of 65 and 117 Is Expressible in the Form 65n – 117, Find n
Step 1: Find the HCF of 65 and 117
Prime factorisation:
- 65 = 5 × 13
- 117 = 3 × 3 × 13
HCF = 13
Step 2: Form the Equation
65n – 117 = 13
65n = 130
n = 2
Answer
The value of n is 2.
Important Question 9: Least Number Which When Divided by 12, 16, and 24 Leaves Remainder 7 in Each Case
Step 1: Recognise the Type of Question
Since the unknown number is being divided by 12, 16, and 24, the required number must be a common multiple of these numbers.
So this is an LCM question.
Step 2: Find the LCM of 12, 16, and 24
Prime factorisation:
- 12 = 2 squared × 3
- 16 = 2 to the power 4
- 24 = 2 cubed × 3
LCM = 2 to the power 4 × 3 = 48
Step 3: Add the Common Remainder
Required number = 48 + 7 = 55
Answer
The least required number is 55.
Important Question 10: Two Positive Numbers Have HCF 12 and Product 6336
Find the number of possible pairs of numbers.
Step 1: Assume the Numbers
If HCF is 12, then the numbers can be written as:
- 12a
- 12b
Step 2: Use the Product Condition
12a × 12b = 6336
144ab = 6336
ab = 44
Step 3: Find Factor Pairs of 44
The factor pairs of 44 are:
- 1 and 44
- 2 and 22
- 4 and 11
Now we keep only those pairs where a and b are co-prime, because the HCF has already been taken out as 12.
- 1 and 44 are co-prime
- 2 and 22 are not co-prime
- 4 and 11 are co-prime
So valid pairs are:
- a = 1, b = 44
- a = 4, b = 11
Step 4: Form the Actual Number Pairs
Using 12a and 12b:
- 12 × 1 = 12 and 12 × 44 = 528
- 12 × 4 = 48 and 12 × 11 = 132
Answer
The number of possible pairs is 2.
The pairs are:
- 12 and 528
- 48 and 132
Short Tricks for Solving Real Numbers Questions Faster
Students should be careful with shortcuts, but some patterns help save time.
Trick for HCF in Remainder Questions
If a number leaves remainders when dividing two or more numbers, subtract the remainders first and then find the HCF.
Trick for LCM in Common Multiple Questions
If the unknown number is divisible by the given numbers, first find the LCM.
If a remainder is left in each case, add that common remainder after finding the LCM.
Trick for LCM in Prime Factorisation
Take the highest powers of all prime factors.
Trick for HCF in Prime Factorisation
Take only the common prime factors with the lowest powers.
Previous Year Questions Pattern Analysis
The previous year questions from Real Numbers often come in these forms:
Board Pattern 1: HCF with Remainders
Students are given two numbers and two remainders, and they must find the greatest divisor.
Board Pattern 2: Assertion and Reason
These questions test conceptual clarity more than calculation.
Board Pattern 3: Ratio and HCF Relationship
Students are given a ratio and the HCF, then asked to find the LCM or actual numbers.
Board Pattern 4: Rational and Irrational Logic
These questions test whether students can identify the correct multiplier or product pattern.
Board Pattern 5: Practical Arrangement Questions
These include room distribution, grouping, divisibility, and equal arrangement problems.
Common Mistakes Students Make in Real Numbers
Confusing HCF with LCM
This is the most common mistake in this chapter.
Forgetting to Adjust Remainders
In questions involving remainders, students often directly find HCF or LCM without first subtracting or adding the remainder as required.
Ignoring the Co-Prime Condition
In pair-based questions, students often include all factor pairs even when only co-prime pairs are valid.
Incorrect Prime Factorisation
One wrong factorisation can spoil the entire answer.
Board Exam Preparation Tips for Real Numbers
Learn the Logic Behind Each Type of Question
Do not only memorise formulas. Understand why HCF or LCM is being used.
Practice Previous Year Questions
Real Numbers is one of the chapters where previous year patterns repeat often.
Write Steps Clearly
Even in short questions, writing clear steps improves accuracy and presentation.
Revise Rational and Irrational Number Basics
Students often focus only on HCF and LCM, but irrational number questions also carry marks.
Practice Questions for Students
Important Practice Questions
- Find the HCF of 96 and 404 using prime factorisation.
- Find the least number that is divisible by 18, 24, and 30.
- Find the greatest number that divides 85 and 155 leaving remainders 1 and 5.
- Find the smallest irrational number by which root 18 must be multiplied to get a rational number.
- Two numbers are in the ratio 3:7 and their HCF is 9. Find their LCM.
FAQs
Q1. How do I know whether to use HCF or LCM in Real Numbers questions?
Use HCF when the question is about greatest common division, equal grouping, or the highest common factor. Use LCM when the question is about the least common multiple, a number divisible by others, or common repetition.
Q2. What is the relationship between HCF and LCM for two numbers?
For any two natural numbers, HCF multiplied by LCM is equal to the product of the two numbers.
Q3. Why do we subtract remainders in some HCF questions?
We subtract the remainders because the required divisor divides the adjusted numbers exactly. That helps us find the correct HCF.
Q4. Why do we add the remainder after finding the LCM in some questions?
In such cases, the unknown number is slightly more than a common multiple and leaves the same remainder when divided by the given numbers.
Q5. How do I find LCM using prime factorisation?
Write the prime factorisation of all numbers and take each prime factor with the highest power appearing among the numbers.
Q6. How do I find HCF using prime factorisation?
Write the prime factorisation of the numbers and take only the common prime factors with the lowest powers.
Q7. What is the easiest way to identify a rational number?
A rational number can be written in the form p/q where q is not equal to zero.
Q8. Can the product of two irrational numbers be rational?
Yes. For example, root 3 and root 27 are both irrational, but their product is 9, which is rational.
Conclusion
Real Numbers is a concept-heavy but highly scoring chapter in Class 10 Maths. Most important questions from this chapter are based on a few repeated patterns such as HCF with remainders, LCM in divisibility questions, ratio-based number formation, and rational-irrational number logic. Students who understand these patterns can solve previous year questions with much greater confidence.
The smartest way to prepare for this chapter is to revise the core concepts first, then practise board-style questions step by step. At Deeksha Vedantu, we believe that when students combine concept clarity with steady practice, chapters like Real Numbers become straightforward and scoring.
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