Short Intro
In this post, students can find complete step-by-step solutions for Class 10 Maths Chapter 1 Real Numbers Exercise 1.3 based on the latest NCERT syllabus. All questions are solved in a simple and exam-oriented format to help students understand Euclid’s Division Lemma and related concepts easily.
Exercise 1.3 is based on Euclid’s Division Algorithm and finding HCF using Euclid’s method.
Quick Information Box
| Item | Details |
|---|---|
| Board | CBSE |
| Class | 10 |
| Subject | Maths |
| Chapter | Real Numbers |
| Exercise | 1.3 |
| Main Topic | Euclid’s Division Lemma |
Concepts Used (Topics Covered)
- Euclid’s Division Lemma
- HCF using Euclid’s Algorithm
- Divisibility
- Prime Numbers
- Rational and Irrational Numbers
The chapter explains divisibility properties and applications of Euclid’s algorithm.
Important Formulas
Euclid’s Division Lemma
a=bq+r, 0≤r<b
HCF Formula
HCF(a,b)=HCF(b,r)
Questions & Step-by-step Solutions
Question 1
Use Euclid’s division algorithm to find the HCF of 135 and 225.
Solution
Using Euclid’s Division Lemma:
Now,
Again,
90=45×2+0
Since remainder becomes 0,
Final Answer:
HCF = 45
Question 2
Use Euclid’s division algorithm to find the HCF of 196 and 38220.
Solution
Applying Euclid’s algorithm:
38220=196×195+0
Since remainder is 0,
Final Answer:
HCF = 196
Question 3
Use Euclid’s division algorithm to find the HCF of 867 and 255.
Solution
Applying Euclid’s division lemma:
867=255×3+102
Now,
255=102×2+51
Again,
102=51×2+0
Final Answer:
HCF = 51
Question 4
A school has received 96 books for Class X and 404 books for Class IX. These books are to be arranged in stacks so that each stack has the same number of books. Find the maximum number of books that each stack can have.
Solution
We need to find:
HCF of 96 and 404
Using Euclid’s algorithm:
404=96×4+20
Now,
96=20×4+16
Again,
20=16×1+4
Now,
16=4×4+0
Final Answer:
Maximum number of books in each stack = 4
Question 5
Find the largest number that divides 70 and 125 leaving remainders 5 and 8 respectively.
Solution
Required divisor:
HCF of (70 − 5) and (125 − 8)
= HCF of 65 and 117
Using Euclid’s algorithm:
117=65×1+52
Now,
65=52×1+13
Again,
52=13×4+0
Final Answer:
Required number = 13
Question 6
Find whether 6ⁿ can end with digit 0 for any natural number n.
Solution
Prime factorisation of:
6n=(2X3)n
A number ending with 0 must contain factor 5.
But:
6ⁿ contains only factors 2 and 3.
Therefore,
Final Answer:
6ⁿ can never end with digit 0.
Common Mistakes
- Writing remainder greater than divisor
- Incorrect division calculations
- Stopping Euclid’s algorithm too early
- Confusing HCF and LCM
Exam Tips
- Always use:
a=bq+r
properly.
- Continue division until remainder becomes 0.
- Write each step clearly for full marks.
- Practice mental division for faster calculations.
Practice MCQs
MCQ 1
Which formula represents Euclid’s Division Lemma?
A. a = b + r
B. a = bq + r
C. a = br
D. a = q + r
Answer:
B. a = bq + r
MCQ 2
The remainder in Euclid’s lemma is always:
A. Greater than divisor
B. Equal to divisor
C. Less than divisor
D. Negative
Answer:
C. Less than divisor
MCQ 3
HCF of 12 and 18 is:
A. 2
B. 3
C. 6
D. 9
Answer:
C. 6
MCQ 4
Which method is used to find HCF quickly?
A. Graph method
B. Euclid’s algorithm
C. Matrix method
D. Elimination method
Answer:
B. Euclid’s algorithm
FAQ Section
What is Euclid’s Division Lemma?
It states that:
a=bq+r
where:
0\leq r<b
Why is Exercise 1.3 important?
It helps students understand HCF calculation and divisibility concepts used in board exams.
Which chapter contains Euclid’s Algorithm?
Chapter 1 – Real Numbers.
Is Euclid’s Algorithm important for competitive exams?
Yes, it is frequently used in aptitude and number system questions.
📘 Improve Your Maths Preparation with MyMockMate!
✅ Practice Chapter-wise Mock Tests
✅ Solve Important MCQs
✅ Download NCERT Notes
✅ Get Instant Performance Analysis
✅ Prepare for Board Exams Smarter
Start learning now on MyMockMate.
About the Author
