Real Numbers Class 10
Master Real Numbers Class 10 with comprehensive NCERT Solutions, Practice Questions, MCQs, Sample Papers, Case Based Questions, and Video lessons.
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NCERT Solutions
Real Numbers Class 10 – NCERT Solutions
Each question below opens its complete step-by-step Teachoo solution.
Ex 1.1
15 questionsEx 1.1, 1 (i)
teackhoo
Ex1.1,1
Express each number as a product of its prime factors:
(i) 140
2|140
2| 70
5} 35
7| 7
1
Hence,
140=2x2x5x7
=22x5x7
Ex 1.1, 1 (ii)
Ex1.1,1 teackhoo
Express each number as a product of its prime factors:
(ii) 156
21/156
2| 78
3] 39
13) 13
1
Hence,
186 =2x2x3x13
=22x3x13
Ex 1.1, 1 (iii)
Ex 4.4, 1 teachoo
Express each number as a product of its prime factors:
(iii) 3825
3] 3825
3} 1275
5S} 425
5 {| 85
17|_ 17
1
Hence,
3825=3x3x5x5x17
= 32x 52x17
Ex 1.1, 1 (iv)
Ex 44,1 teachoo
Express each number as a product of its prime factors:
(iv) 5005
5|5005
7 {1001
11} 143
13] 13
1
Hence,
5005 =5x7x11x 13
Ex 1.1, 1 (v)
Ex1.1,1 teachoo
Express each number as a product of its prime factors:
(v) 7429
17|7429
19] 437
23] 23
1
Hence,
7429 = 17x 19 x 23
Ex 1.1, 2 (i)
teackhoo
Ex 1.1, 2
Find the LCM and HCF of the following pairs of integers and
verify that LCM x HCF = product of the two numbers.
(i) 26 and 91
Finding HCF
2 [26 7/91
13] 13 13/13
1 1
26 = 2 x|13
91=7 x13
“ H.C.F = 13
Ex 1.1, 2 (ii)
Ex 1.1, 2 teackhoo
Find the LCM and HCF of the following pairs of integers and verify
that LCM x HCF = product of the two numbers.
(ii) 510 and 92
Finding HCF
2 |510 2 192
3 [255 2 |46
5 | 85 23 |23
17| 17 1
1
510 =|2k 3x5%x17
92 =|2K 2 x 23
* H.C = 2
Ex 1.1, 2 (iii)
Ex 1.1, 2 teackoo
Find the LCM and HCF of the following pairs of integers and verify
that LCM x HCF = product of the two numbers.
(iii) 336 and 54
Finding HCF
2 [336
2_ [168 2 [54
2 | 84 3 27
2 | 42 3|9
3] 21 313
7| 7 1
1
336 =|2|x 2x 2x 2 xI3ix 7
54 =|2)k 3x3 x!
* H.C.F = 2x 3
=6
Ex 1.1, 3 (i)
Ex 1.13 teackhoo
Find the LCM and HCF of the following integers by applying the
prime factorisation method.
(i) 12,15 and 21
Finding HCF
2 {12 3 [45 3| 24
2/6 5|5 7/7
3
3 1 1
1
12=2x2 (3
15 =|3]x 5
21 =|3|x 7
- H.C.F=3
Ex 1.1, 3 (ii)
Ex 1.1, 3 teackhoo
Find the LCM and HCF of the following integers by applying the prime
factorisation method.
(ii) 17, 23 and 29
Finding HCF
17| 17 23) 23 29/29
1 1 1
17=17
23 = 23
29=29
Since no factor is common
So, 1 must be common
~ HCF=1
Ex 1.1, 3 (iii)
Ex 1.1, 3 teackhoo
Find the LCM and HCF of the following integers by applying the
prime factorisation method.
(iii) 8, 9 and 25
Finding HCF
218 3/9 5 [25
2\4 3|3 5] 5
2/2 1 1
1
8=2x2x2
9=3x3
25=5x5
Since no factor is common
So, 1 must be common
« H.CF=1
Ex 1.1, 4
ex4.1,4 teackhoo
Given that HCF (306, 657) = 9, find LCM (306, 657}.
We know that
H.C.F x L.C.M = Product of numbers
9xL.C.M = 306 x 657
Lom = 306% 697
9
L.C.M = 22338
- L.C.M of 306 & 657 is 22338
Ex 1.1, 5
Ex 1.1, 5 teackoo
Check whether 6" can end with the digit 0 for any natural number n.
Let us take the example of a number which ends with the digit 0
So,
10=2x5
100 =2x2x5x5
Here we note that numbers ending with 0 has both 2 and 5 as their
prime factors
Whereas
6" = (2 x 3)"
Does not have 5 as a prime factor.
Ex 1.1, 6
Ex 1.1, 6 teachoo
Explain why 7x 11x 13+13and7x6x5x4x3x2x1+S5are
composite numbers.
Checking 7 x 11 x 13 + 13
7x11x«13+13
=13x (7x11 +1)
=13 x (77+1}
=13x78
=13x13x3x2
Since it has more than two factors
(13, 3, 2, 1,13 x 78),
« [tis a composite number
Ex 1.1, 7
In video, the answer should be 36 not 54
View solutionEx 1.2
5 questionsEx 1.2, 1
teachoo
Ex 1.2, 1
Prove that V5 is irrational.
We have to prove 5 is irrational
Let us assume the opposite,
i.e., V5 is rational
Hence, V5 can be written in the form .
where a and b (b# 0) are co-prime (no common factor other than 1)
Hence,
a
v5=¢
V5b=a
Squaring both sides
(V5b)? = a?
Ex 1.2, 2
teackhoo
Ex 1.2, 2
Prove that 3 + 2V5 is irrational.
We have to prove 3 + 2V5 is irrational
Let us assume the opposite,
ie, 3+ 2V5 is rational
Hence, 3 + 2¥5 can be written in the form ;
where a and b (bz 0) are co-prime (no common factor other than 1)
Hence,
3+2V5 =>
2v5=5-3
a —3b
aye =4=*
Ex 1.2, 3 (i)
Ex 1.2, 3 teackhoo
Prove that the following are irrationals :
5 1
OF
We have to prove ra is irrational
Let us assume the opposite,
ie. = is rational
Hence, 4 can be written in the form =
v2 b
where a and b (bz 0) are co-prime (no common factor other than 1)
Hence,
1.2
v2 »B
b= V2
a
Ex 1.2, 3 (ii)
Ex 1.2, 3 teackoo
Prove that the following are irrationals :
(ii) 7V5
We have to prove 775 is irrational
Let us assume the opposite,
i.e., 7V5 is rational
Hence, 7\V5 can be written in the form ;
where a and b (bz 0) are co-prime (no common factor other than 1)
Hence,
75 ="
b
1 a
B= 5% 5
Ex 1.2, 3 (iii)
Ex 1.2, 3 teackoo
Prove that the following are irrationals :
(ii) 6+ V2
We have to prove 6 + V2 is irrational
Let us assume the opposite,
ie., 6+ V2 is rational
Hence, 6 + ¥2 can be written in the form ©
where a and b (bz 0) are co-prime (no common factor other than 1)
Hence,
6+V2= ;
a
V2 =57 6
Examples
12 questionsExample 1
Example 1 teachoo
Consider the numbers 4", where n is a natural number. Check
whether there is any value of n for which 4" ends with the digit zero.
Let us take the example of a number which ends with the digit 0
So,
10=2x5
100=2x2x5x5
Here we note that numbers ending with 0 has both 2 and 5 as their
prime factors
Whereas
4r=(2« 2)"
Example 2
teachoo
Example 2
Find the LCM and HCF of 6 and 20 by the prime factorisation method.
Finding HCF
2|6 2{20
3|3 2{10
1 5|5
1
6 =|2k3
20=[2k 2x 5S
* HC.F = 2
Example 3
Example 3 teackoo
Find the HCF of 96 and 404 by the prime factorisation method.
Hence, find their LCM
Finding HCF
2|96
2 [48 2 1404
2124 21202
2112 101} 101
2/16 1
3/3
1
96 =|2%2k2x2x2x3
404 =|2*2 k 101
H.C.F = 2x2
=4
Example 4
Example 4 teackoo
Find the HCF and LCM of 6, 72 and 120, using the prime
factorisation method.
2|6 2|72 2 [120
3/3 2/36 2160
1 2/18 2130
319 3] 15
313 Sts
1 1
6 =|2| x3
72 =2)x 2 x 2 x13]x 3
120 x2x2>43)x5
H.C.F = 2x3
=6
Example 5
teachoo
Example 5
Prove that V3 is irrational.
We have to prove 73 is irrational
Let us assume the opposite,
i.e., V3 is rational
Hence, ¥3 can be written in the form .
where a and b (b# 0) are co-prime (no common factor other than 1)
Hence,
a
Be!
V3b=a
Squaring both sides
(V3b)2 = a2
Example 6
teackhoo
Example 6
Show that 5 - V3 is irrational.
We have to prove 5 - V3 is irrational
Let us assume the opposite,
i.e, 5- ¥3 is rational
Hence, 5 - ¥3 can be written in the form ©
where a and b (bz 0) are co-prime (no common factor other than 1)
Hence,
a
5- V3= b
a
—v3 =a 5
a-—5b
-V3= <=
Example 7
teackhoo
Example 7
Show that 3V2 is irrational.
We have to prove 3y2 is irrational
Let us assume the opposite,
ie., 3V72 is rational
Hence, 3V2 can be written in the form ;
where a and b (bz 0) are co-prime (no common factor other than 1)
Hence,
a
3V2 =>
1 a
V2=5%5
a
v2= 35
Prove that root 2 is Irrational
Theorem 10.4
Prove that √2 is irrational.
Question 1
teackoo.com
Example 1
Use Euclid’s algorithm to find the HCF of 4052 and 12576.
Since 12576 > 4052,
We divide 12576 by 4052
4052 hros7e\ 3
12156
qe
420
Since remainder is not 0
We divide 4052 by 420
Question 2
feackoo.com
Example 2
Show that every positive even integer is of the form 2q, and that
every positive odd integer is of the form 2q+ 1, where q is some
integer.
As per Euclid’s Division Lemma
If a and b are 2 positive integers, then
a=bq+tr
where O<r<b
Let positive integer be a
And b=2
Hence a=2q+r
where (0 <r < 2}
ris an integer greater than or equal to 0 and less than 2
hence r can be either 0 or 1
Question 3
feackoo.com
Example 3
Show that any positive odd integer is of the form 4q+ 1 or 4q+ 3,
where q is some integer.
As per Euclid’s Division Lemma
If a and b are 2 positive integers, then
a=bqtr
where O<r<b
Let positive integer be a
And b=4
Hence a=4q+r
where (0 <r <4)
ris an integer greater than or equal to 0 and less than 4
hence r can be either0,1,2o0r3
Question 4
teackoo.com
Example 4 (Introduction)
A sweet seller has 420 kaju barfis and 130 badam barfis. She wants
to stack them in such a way that each stack has the same number,
and they take up the least area of the tray. What is the maximum
number of barfis that can be placed in each stack for this purpose?
Suppose we have to stack 12 kaju barfis & 4 badam barfis
oooe ee °
Oooaao ee °
oo °
oooda e
ooada Od o
oO
OO
Maximum number of barfis to stack = 4 O
=HCF of 12 &4
Similarly , we will do in this question
MCQs from NCERT Exemplar
12 questionsQuestion 1
The decimal expansion of the rational number 33/(2^2 5) will terminate after
(A)one decimal place
(B) two decimal places
(C) three decimal places
(D) more than 3 decimal places
Question 2
Euclid’s division lemma states that for two positive integers a
and b, there exist unique integers q and r such that a = bq + r, where r must satisfy
(A) 1 < r < b
(B) 0 < r ≤ b
(C) 0 ≤ r < b
(D) 0 < r < b
Euclid’s Division Lemma states that
Given positive integers a and b,
there exist unique integers q and r satisfying
a = bq + r,
where
0 ≤ r < b
So, correct
answer is (C)
Question 3
For some integer m, every even integer is of the form:
(a) m
(b) m + 1
(c) 2m
(d) 2m + 1
For more details, check
Example 2 - Chapter 1 Class 10
Question 4
For some integer q, every odd integer is of the form:
(a)q
(b) q + 1
(c) 2q
(d) 2q + 1
For more details, check
Example 2 - Chapter 1 Class 10
Question 5
n
2
– 1 is divisible by 8, if n is:
(a)an integer
(b) a natural number
(c) an odd integer
(d) an even integer
Question 6
If the HCF of 65 and 117 is expressible in the form 65m – 117, then the value of m is
(A) 4
(B) 2
(C) 1
(D) 3
Question 7
The largest number which divides 70 and 125, leaving remainders 5 and 8, respectively, is:
(a)13
(b) 65
(c) 875
(d) 1,750
Question 8
If two positive integers a and b are written as a = x
3
y2 and b = xy
3
; x, y are prime numbers, then HCF (a, b) is:
(a)xy
(b) xy
2
(c) x
3
y
3
(d) x2 y
2
Question 9
If two positive integers p and q can be expressed as
p = ab
2
and q = a
3
b; a, b being prime numbers, then LCM (p, q) is
(A) ab
(B) a
2
b
2
(C) a
3
b
2
(D) a
3
b
3
Question 10
The product of a non-zero rational and an irrational number is:
(a) always irrational
(b) always rational
(c) rational or irrational
(d) one
Question 11
The least number that is divisible by all the numbers from 1 to 10 (both inclusive) is
(A) 10
(B) 100
(C) 504
(D) 2520
Question 12
The decimal expansion of the rational number 14587/1250 will terminate after:
(a)one decimal place
(b) two decimal places
(c) three decimal places
(d) four decimal places
Why Learn This With Teachoo?
Real Numbers is Chapter 1 of NCERT Class 10 Mathematics. It develops Euclid’s division algorithm, the Fundamental Theorem of Arithmetic, HCF and LCM, irrational-number proofs and decimal expansions of rational numbers. Teachoo provides step-by-step NCERT solutions, examples, concept lessons, case-based questions, exemplar MCQs and past-year questions for complete Class 10 Real Numbers preparation.
What do you learn in Real Numbers?
Euclid’s division lemma states that for positive integers a and b, there exist unique integers q and r such that a = bq + r, where 0 ≤ r < b. Repeated application produces Euclid’s division algorithm for finding the HCF. The same structure can prove divisibility statements and properties of integers.
The Fundamental Theorem of Arithmetic says that every composite number has a unique prime factorisation apart from the order of factors. Prime factorisation is used to find HCF and LCM and to prove that certain numbers cannot have a proposed form.
Students also prove irrationality, commonly using contradiction. To show √2 is irrational, one assumes it equals p/q in lowest terms and derives that both p and q must be even, contradicting the lowest-term condition.
For a rational number p/q in lowest form, its decimal expansion terminates exactly when q has only 2 and 5 as prime factors. Otherwise the decimal is non-terminating recurring. This gives a factorisation-based test without performing long division.
Topics available on Teachoo
-
Exercises 1.1 and 1.2 and NCERT examples;
-
Euclid’s division algorithm for HCF;
-
proofs using Euclid’s algorithm;
-
prime factorisation;
-
HCF and LCM;
-
decimal expansions;
-
irrational-number proofs;
-
case-based questions and NCERT Exemplar MCQs; and
-
past-year MCQs for Maths Standard.
Learning outcomes
Students should be able to apply Euclid’s algorithm, find HCF and LCM through prime factors, use the HCF–LCM relation appropriately and predict the decimal form of a rational number. They should construct a valid contradiction proof for irrationality and explain the condition attached to every theorem used.
Why is this chapter important?
Real Numbers supports algebra, number theory, decimal representation and proof. It is a frequent source of short-answer, assertion-reasoning and case-based board questions because a small calculation must be combined with a precise theorem.
How Teachoo helps you prepare
Learn the concepts before opening the exercise answers. In Euclid questions, write every division step until the remainder becomes zero; the last non-zero remainder is the HCF. In decimal-expansion questions, reduce the fraction before examining the denominator. For proofs, state the assumption and contradiction explicitly.
Teachoo lets students revise serial-order NCERT solutions or study concept-wise from easy to difficult. After textbook questions, use exemplar, case-based and past-year MCQs to test recognition under exam conditions.
Board-exam and competency preparation
Real Numbers questions often present the theorem indirectly. A case may ask when groups repeat together, whether a decimal terminates, or why a proposed square root cannot be rational. Begin by naming the governing idea—Euclid’s algorithm, prime factorisation, denominator test or contradiction—before calculating. In proof questions, each implication must be justified; examples cannot establish a statement about every integer.
For MCQs, eliminate options using factor structure before doing long calculations. For assertion-reasoning questions, judge the assertion and reason separately, then decide whether the reason actually explains the assertion. Keep exact values until the conclusion and distinguish “irrational” from merely “non-terminating.”
Quick revision checklist
Find one HCF by Euclid’s algorithm and by prime factorisation, verify an HCF–LCM result, classify several decimal expansions from reduced denominators and write one complete irrationality proof. Finish with a case-based question and check that every theorem condition has been stated.
Common mistakes to avoid
Do not apply the terminating-decimal test before reducing p/q. Euclid’s remainder must satisfy 0 ≤ r < divisor. The identity HCF × LCM = product applies directly to a pair of positive integers, not an arbitrary list. In irrationality proofs, do not merely state that a number is irrational; show why the rational assumption fails.
Deeper reasoning and concept connections
A student has understood Real Numbers only when the idea can be moved between words, diagrams, examples and mathematical notation. Start with a concrete example, identify what changes and what remains fixed, represent the relationship clearly and then state the rule. This movement between representations is important because school and competency questions often present a familiar idea in an unfamiliar form.
The chapter should also be connected to earlier and later mathematics. Definitions supply the language, worked examples reveal the method, and mixed questions test whether the method can be selected without a hint. Instead of memorising the appearance of a solved question, ask what information triggered the method, which condition made it valid and how the answer could be checked. That makes learning transferable to later chapters rather than limited to one exercise.
How to solve unfamiliar and competency-based questions
Read the complete problem before calculating. Underline the quantities, conditions and command word—find, compare, construct, justify, estimate or prove. Rephrase the task in one sentence and choose a representation such as a table, labelled figure, number line, expression or graph. Solve in small steps, keeping units and labels visible.
For an application question, the final line must answer the situation, not only display a number. For an assertion–reason question, test the assertion and reason separately before deciding whether one explains the other. For an MCQ, eliminate options using definitions, signs, size estimates or boundary cases before performing long calculations. If the answer is visual, check it against the stated scale or construction conditions rather than the appearance of the drawing.
What complete mastery looks like
For Real Numbers, a student should be able to define the central ideas in simple language, recognise them in different representations, solve routine questions accurately and explain the method used. They should also be able to correct a flawed solution, create an example satisfying given conditions and combine two ideas from the chapter in one problem. A reliable mastery test is to solve one direct question, one application question and one reasoning question without looking at notes, then explain all three aloud or in writing.
Keep a compact error log with four labels: concept, interpretation, calculation and presentation. Reattempt each error after a gap instead of rereading the answer immediately. Improvement comes from correcting the decision that caused the mistake, not from repeating questions whose method is already known.
Additional frequently asked questions
What should a student know before starting Real Numbers?
Revise the definitions, number operations, diagrams or notation used at the beginning of the chapter. The prerequisite list should be short: if an earlier skill blocks progress, repair that skill with two or three focused questions and return to the chapter.
How can a student check an answer in Real Numbers?
Use an independent check whenever possible: substitute the result, reverse the operation, estimate its size, compare it with the figure, test a simpler case or solve using another representation. A check should examine the mathematical condition, not merely repeat the same arithmetic.
How many questions are enough for strong preparation?
There is no fixed number. Stop counting questions and track coverage: every concept, every standard method, at least one mixed problem, one competency-based problem and every previously incorrect type should be solved independently. Ten varied, analysed questions are more valuable than fifty copied solutions.
How should Teachoo solutions be used without becoming dependent on them?
Attempt the question first and mark the exact step where progress stops. Read only enough of the solution to repair that step, close it and restart the question. Finally, solve a similar problem without help. This turns a solution into feedback rather than a substitute for thinking.
Frequently asked questions
What is the main purpose of Euclid’s division algorithm?
It provides an efficient repeated-division method for finding the HCF of two positive integers.
When does a rational number have a terminating decimal?
When its reduced denominator has no prime factors other than 2 and 5.
Does Teachoo include board-focused practice?
Yes. The chapter includes case-based questions, NCERT Exemplar MCQs and past-year Maths Standard MCQs alongside NCERT solutions.
Master theorem conditions, not just formulas. Real Numbers rewards precise reasoning more than mechanical memorisation.