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# NCERT Textbook - Real Numbers Class 10 Notes | EduRev

## Class 10 : NCERT Textbook - Real Numbers Class 10 Notes | EduRev

``` Page 1

REAL NUMBERS 1
1
1.1 Introduction
In Class IX, you began your exploration of the world of real numbers and encountered
irrational numbers. We continue our discussion on real numbers in this chapter. We
begin with two very important properties of positive integers in Sections 1.2 and 1.3,
namely the Euclid’s division algorithm and the Fundamental Theorem of Arithmetic.
Euclid’s division algorithm, as the name suggests, has to do with divisibility of
integers. Stated simply, it says any positive integer a can be divided by another positive
integer b in such a way that it leaves a remainder r that is smaller than b. Many of you
probably recognise this as the usual long division process. Although this result is quite
easy to state and understand, it has many applications related to the divisibility properties
of integers. We touch upon a few of them, and use it mainly to compute the HCF of
two positive integers.
The Fundamental Theorem of Arithmetic, on the other hand, has to do something
with multiplication of positive integers. You already know that every composite number
can be expressed as a product of primes in a unique way— this important fact is the
Fundamental Theorem of Arithmetic. Again, while it is a result that is easy to state and
understand, it has some very deep and significant applications in the field of mathematics.
We use the Fundamental Theorem of Arithmetic for two main applications. First, we
use it to prove the irrationality of many of the numbers you studied in Class IX, such as
2 , 3 and
5
. Second, we apply this theorem to explore when exactly the decimal
expansion of a rational number, say ( 0)
p
q
q
? , is terminating and when it is non-
terminating repeating. W e do so by looking at the prime factorisation of the denominator
q of
p
q
. Y ou will see that the prime factorisation of q will completely reveal the nature
of the decimal expansion of
p
q
.
So let us begin our exploration.
REAL NUMBERS
2020-21
Page 2

REAL NUMBERS 1
1
1.1 Introduction
In Class IX, you began your exploration of the world of real numbers and encountered
irrational numbers. We continue our discussion on real numbers in this chapter. We
begin with two very important properties of positive integers in Sections 1.2 and 1.3,
namely the Euclid’s division algorithm and the Fundamental Theorem of Arithmetic.
Euclid’s division algorithm, as the name suggests, has to do with divisibility of
integers. Stated simply, it says any positive integer a can be divided by another positive
integer b in such a way that it leaves a remainder r that is smaller than b. Many of you
probably recognise this as the usual long division process. Although this result is quite
easy to state and understand, it has many applications related to the divisibility properties
of integers. We touch upon a few of them, and use it mainly to compute the HCF of
two positive integers.
The Fundamental Theorem of Arithmetic, on the other hand, has to do something
with multiplication of positive integers. You already know that every composite number
can be expressed as a product of primes in a unique way— this important fact is the
Fundamental Theorem of Arithmetic. Again, while it is a result that is easy to state and
understand, it has some very deep and significant applications in the field of mathematics.
We use the Fundamental Theorem of Arithmetic for two main applications. First, we
use it to prove the irrationality of many of the numbers you studied in Class IX, such as
2 , 3 and
5
. Second, we apply this theorem to explore when exactly the decimal
expansion of a rational number, say ( 0)
p
q
q
? , is terminating and when it is non-
terminating repeating. W e do so by looking at the prime factorisation of the denominator
q of
p
q
. Y ou will see that the prime factorisation of q will completely reveal the nature
of the decimal expansion of
p
q
.
So let us begin our exploration.
REAL NUMBERS
2020-21
2 MATHEMA TICS
1.2 Euclid’s Division Lemma
Consider the following folk puzzle*.
A trader was moving along a road selling eggs. An idler who didn’t have
much work to do, started to get the trader into a wordy duel. This grew into a
fight, he pulled the basket with eggs and dashed it on the floor. The eggs broke.
The trader requested the Panchayat to ask the idler to pay for the broken eggs.
The Panchayat asked the trader how many eggs were broken. He gave the
following response:
If counted in pairs, one will remain;
If counted in threes, two will remain;
If counted in fours, three will remain;
If counted in fives, four will remain;
If counted in sixes, five will remain;
If counted in sevens, nothing will remain;
My basket cannot accomodate more than 150 eggs.
So, how many eggs were there? Let us try and solve the puzzle. Let the number
of eggs be a. Then working backwards, we see that a is less than or equal to 150:
If counted in sevens, nothing will remain, which translates to a = 7p + 0, for
some natural number p. If counted in sixes, a = 6q+ 5, for some natural number q.
If counted in fives, four will remain. It translates to a = 5w + 4, for some natural
number w.
If counted in fours, three will remain. It translates to a = 4s + 3, for some natural
number s.
If counted in threes, two will remain. It translates to a = 3t + 2, for some natural
number t.
If counted in pairs, one will remain. It translates to a = 2u + 1, for some natural
number u.
That is, in each case, we have a and a positive integer b (in our example,
b takes values 7, 6, 5, 4, 3 and 2, respectively) which divides a and leaves a remainder
r (in our case, r is 0, 5, 4, 3, 2 and 1, respectively), that is smaller than b. The
* This is modified form of a puzzle given in ‘Numeracy Counts!’ by A. Rampal, and others.
2020-21
Page 3

REAL NUMBERS 1
1
1.1 Introduction
In Class IX, you began your exploration of the world of real numbers and encountered
irrational numbers. We continue our discussion on real numbers in this chapter. We
begin with two very important properties of positive integers in Sections 1.2 and 1.3,
namely the Euclid’s division algorithm and the Fundamental Theorem of Arithmetic.
Euclid’s division algorithm, as the name suggests, has to do with divisibility of
integers. Stated simply, it says any positive integer a can be divided by another positive
integer b in such a way that it leaves a remainder r that is smaller than b. Many of you
probably recognise this as the usual long division process. Although this result is quite
easy to state and understand, it has many applications related to the divisibility properties
of integers. We touch upon a few of them, and use it mainly to compute the HCF of
two positive integers.
The Fundamental Theorem of Arithmetic, on the other hand, has to do something
with multiplication of positive integers. You already know that every composite number
can be expressed as a product of primes in a unique way— this important fact is the
Fundamental Theorem of Arithmetic. Again, while it is a result that is easy to state and
understand, it has some very deep and significant applications in the field of mathematics.
We use the Fundamental Theorem of Arithmetic for two main applications. First, we
use it to prove the irrationality of many of the numbers you studied in Class IX, such as
2 , 3 and
5
. Second, we apply this theorem to explore when exactly the decimal
expansion of a rational number, say ( 0)
p
q
q
? , is terminating and when it is non-
terminating repeating. W e do so by looking at the prime factorisation of the denominator
q of
p
q
. Y ou will see that the prime factorisation of q will completely reveal the nature
of the decimal expansion of
p
q
.
So let us begin our exploration.
REAL NUMBERS
2020-21
2 MATHEMA TICS
1.2 Euclid’s Division Lemma
Consider the following folk puzzle*.
A trader was moving along a road selling eggs. An idler who didn’t have
much work to do, started to get the trader into a wordy duel. This grew into a
fight, he pulled the basket with eggs and dashed it on the floor. The eggs broke.
The trader requested the Panchayat to ask the idler to pay for the broken eggs.
The Panchayat asked the trader how many eggs were broken. He gave the
following response:
If counted in pairs, one will remain;
If counted in threes, two will remain;
If counted in fours, three will remain;
If counted in fives, four will remain;
If counted in sixes, five will remain;
If counted in sevens, nothing will remain;
My basket cannot accomodate more than 150 eggs.
So, how many eggs were there? Let us try and solve the puzzle. Let the number
of eggs be a. Then working backwards, we see that a is less than or equal to 150:
If counted in sevens, nothing will remain, which translates to a = 7p + 0, for
some natural number p. If counted in sixes, a = 6q+ 5, for some natural number q.
If counted in fives, four will remain. It translates to a = 5w + 4, for some natural
number w.
If counted in fours, three will remain. It translates to a = 4s + 3, for some natural
number s.
If counted in threes, two will remain. It translates to a = 3t + 2, for some natural
number t.
If counted in pairs, one will remain. It translates to a = 2u + 1, for some natural
number u.
That is, in each case, we have a and a positive integer b (in our example,
b takes values 7, 6, 5, 4, 3 and 2, respectively) which divides a and leaves a remainder
r (in our case, r is 0, 5, 4, 3, 2 and 1, respectively), that is smaller than b. The
* This is modified form of a puzzle given in ‘Numeracy Counts!’ by A. Rampal, and others.
2020-21
REAL NUMBERS 3
moment we write down such equations we are using Euclid’s division lemma,
which is given in Theorem 1.1.
Getting back to our puzzle, do you have any idea how you will solve it? Yes! Y ou
must look for the multiples of 7 which satisfy all the conditions. By trial and error
(using the concept of LCM), you will find he had 119 eggs.
In order to get a feel for what Euclid’s division lemma is, consider the following
pairs of integers:
17, 6; 5, 12; 20, 4
Like we did in the example, we can write the following relations for each such
pair:
17 = 6 × 2 + 5 (6 goes into 17 twice and leaves a remainder 5)
5 = 12 × 0 + 5 (This relation holds since 12 is larger than 5)
20 = 4 × 5 + 0 (Here 4 goes into 20 five-times and leaves no remainder)
That is, for each pair of positive integers a and b, we have found whole numbers
q and r, satisfying the relation:
a = bq + r, 0 = r < b
Note that q or r can also be zero.
Why don’t you now try finding integers q and r for the following pairs of positive
integers a and b?
(i) 10, 3; (ii) 4, 19; (iii) 81, 3
Did you notice that q and r are unique? These are the only integers satisfying the
conditions a = bq + r, where 0 = r < b. You may have also realised that this is nothing
but a restatement of the long division process you have been doing all these years, and
that the integers q and r are called the quotient and remainder, respectively.
A formal statement of this result is as follows :
Theorem 1.1 (Euclid’s Division Lemma) : Given positive integers a and b,
there exist unique integers q and r satisfying a = bq + r, 0 = r < b.
This result was perhaps known for a long time, but was first recorded in Book VII
of Euclid’s Elements. Euclid’s division algorithm is based on this lemma.
2020-21
Page 4

REAL NUMBERS 1
1
1.1 Introduction
In Class IX, you began your exploration of the world of real numbers and encountered
irrational numbers. We continue our discussion on real numbers in this chapter. We
begin with two very important properties of positive integers in Sections 1.2 and 1.3,
namely the Euclid’s division algorithm and the Fundamental Theorem of Arithmetic.
Euclid’s division algorithm, as the name suggests, has to do with divisibility of
integers. Stated simply, it says any positive integer a can be divided by another positive
integer b in such a way that it leaves a remainder r that is smaller than b. Many of you
probably recognise this as the usual long division process. Although this result is quite
easy to state and understand, it has many applications related to the divisibility properties
of integers. We touch upon a few of them, and use it mainly to compute the HCF of
two positive integers.
The Fundamental Theorem of Arithmetic, on the other hand, has to do something
with multiplication of positive integers. You already know that every composite number
can be expressed as a product of primes in a unique way— this important fact is the
Fundamental Theorem of Arithmetic. Again, while it is a result that is easy to state and
understand, it has some very deep and significant applications in the field of mathematics.
We use the Fundamental Theorem of Arithmetic for two main applications. First, we
use it to prove the irrationality of many of the numbers you studied in Class IX, such as
2 , 3 and
5
. Second, we apply this theorem to explore when exactly the decimal
expansion of a rational number, say ( 0)
p
q
q
? , is terminating and when it is non-
terminating repeating. W e do so by looking at the prime factorisation of the denominator
q of
p
q
. Y ou will see that the prime factorisation of q will completely reveal the nature
of the decimal expansion of
p
q
.
So let us begin our exploration.
REAL NUMBERS
2020-21
2 MATHEMA TICS
1.2 Euclid’s Division Lemma
Consider the following folk puzzle*.
A trader was moving along a road selling eggs. An idler who didn’t have
much work to do, started to get the trader into a wordy duel. This grew into a
fight, he pulled the basket with eggs and dashed it on the floor. The eggs broke.
The trader requested the Panchayat to ask the idler to pay for the broken eggs.
The Panchayat asked the trader how many eggs were broken. He gave the
following response:
If counted in pairs, one will remain;
If counted in threes, two will remain;
If counted in fours, three will remain;
If counted in fives, four will remain;
If counted in sixes, five will remain;
If counted in sevens, nothing will remain;
My basket cannot accomodate more than 150 eggs.
So, how many eggs were there? Let us try and solve the puzzle. Let the number
of eggs be a. Then working backwards, we see that a is less than or equal to 150:
If counted in sevens, nothing will remain, which translates to a = 7p + 0, for
some natural number p. If counted in sixes, a = 6q+ 5, for some natural number q.
If counted in fives, four will remain. It translates to a = 5w + 4, for some natural
number w.
If counted in fours, three will remain. It translates to a = 4s + 3, for some natural
number s.
If counted in threes, two will remain. It translates to a = 3t + 2, for some natural
number t.
If counted in pairs, one will remain. It translates to a = 2u + 1, for some natural
number u.
That is, in each case, we have a and a positive integer b (in our example,
b takes values 7, 6, 5, 4, 3 and 2, respectively) which divides a and leaves a remainder
r (in our case, r is 0, 5, 4, 3, 2 and 1, respectively), that is smaller than b. The
* This is modified form of a puzzle given in ‘Numeracy Counts!’ by A. Rampal, and others.
2020-21
REAL NUMBERS 3
moment we write down such equations we are using Euclid’s division lemma,
which is given in Theorem 1.1.
Getting back to our puzzle, do you have any idea how you will solve it? Yes! Y ou
must look for the multiples of 7 which satisfy all the conditions. By trial and error
(using the concept of LCM), you will find he had 119 eggs.
In order to get a feel for what Euclid’s division lemma is, consider the following
pairs of integers:
17, 6; 5, 12; 20, 4
Like we did in the example, we can write the following relations for each such
pair:
17 = 6 × 2 + 5 (6 goes into 17 twice and leaves a remainder 5)
5 = 12 × 0 + 5 (This relation holds since 12 is larger than 5)
20 = 4 × 5 + 0 (Here 4 goes into 20 five-times and leaves no remainder)
That is, for each pair of positive integers a and b, we have found whole numbers
q and r, satisfying the relation:
a = bq + r, 0 = r < b
Note that q or r can also be zero.
Why don’t you now try finding integers q and r for the following pairs of positive
integers a and b?
(i) 10, 3; (ii) 4, 19; (iii) 81, 3
Did you notice that q and r are unique? These are the only integers satisfying the
conditions a = bq + r, where 0 = r < b. You may have also realised that this is nothing
but a restatement of the long division process you have been doing all these years, and
that the integers q and r are called the quotient and remainder, respectively.
A formal statement of this result is as follows :
Theorem 1.1 (Euclid’s Division Lemma) : Given positive integers a and b,
there exist unique integers q and r satisfying a = bq + r, 0 = r < b.
This result was perhaps known for a long time, but was first recorded in Book VII
of Euclid’s Elements. Euclid’s division algorithm is based on this lemma.
2020-21
4 MATHEMA TICS
An algorithm is a series of well defined steps
which gives a procedure for solving a type of
problem.
The word algorithm comes from the name
of the 9th century Persian mathematician
al-Khwarizmi. In fact, even the word ‘algebra’
is derived from a book, he wrote, called Hisab
al-jabr w’al-muqabala.
A lemma is a proven statement used for
proving another statement.
Euclid’s division algorithm is a technique to compute the Highest Common Factor
(HCF) of two given positive integers. Recall that the HCF of two positive integers a
and b is the largest positive integer d that divides both a and b.
Let us see how the algorithm works, through an example first. Suppose we need
to find the HCF of the integers 455 and 42. We start with the larger integer, that is,
455. Then we use Euclid’s lemma to get
455 = 42 × 10 + 35
Now consider the divisor 42 and the remainder 35, and apply the division lemma
to get
42 = 35 × 1 + 7
Now consider the divisor 35 and the remainder 7, and apply the division lemma
to get
35 = 7 × 5 + 0
Notice that the remainder has become zero, and we cannot proceed any further.
We claim that the HCF of 455 and 42 is the divisor at this stage, i.e., 7. You can easily
verify this by listing all the factors of 455 and 42. Why does this method work? It
works because of the following result.
So, let us state Euclid’s division algorithm clearly.
To obtain the HCF of two positive integers, say c and d, with c > d, follow
the steps below:
Step 1 : Apply Euclid’s division lemma, to c and d. So, we find whole numbers, q and
r such that c = dq + r, 0 = r < d.
Step 2 : If r = 0, d is the HCF of c and d. If r ? 0, apply the division lemma to d and r.
Step 3 : Continue the process till the remainder is zero. The divisor at this stage will
be the required HCF.
Muhammad ibn Musa al-Khwarizmi
(C.E. 780 – 850)
2020-21
Page 5

REAL NUMBERS 1
1
1.1 Introduction
In Class IX, you began your exploration of the world of real numbers and encountered
irrational numbers. We continue our discussion on real numbers in this chapter. We
begin with two very important properties of positive integers in Sections 1.2 and 1.3,
namely the Euclid’s division algorithm and the Fundamental Theorem of Arithmetic.
Euclid’s division algorithm, as the name suggests, has to do with divisibility of
integers. Stated simply, it says any positive integer a can be divided by another positive
integer b in such a way that it leaves a remainder r that is smaller than b. Many of you
probably recognise this as the usual long division process. Although this result is quite
easy to state and understand, it has many applications related to the divisibility properties
of integers. We touch upon a few of them, and use it mainly to compute the HCF of
two positive integers.
The Fundamental Theorem of Arithmetic, on the other hand, has to do something
with multiplication of positive integers. You already know that every composite number
can be expressed as a product of primes in a unique way— this important fact is the
Fundamental Theorem of Arithmetic. Again, while it is a result that is easy to state and
understand, it has some very deep and significant applications in the field of mathematics.
We use the Fundamental Theorem of Arithmetic for two main applications. First, we
use it to prove the irrationality of many of the numbers you studied in Class IX, such as
2 , 3 and
5
. Second, we apply this theorem to explore when exactly the decimal
expansion of a rational number, say ( 0)
p
q
q
? , is terminating and when it is non-
terminating repeating. W e do so by looking at the prime factorisation of the denominator
q of
p
q
. Y ou will see that the prime factorisation of q will completely reveal the nature
of the decimal expansion of
p
q
.
So let us begin our exploration.
REAL NUMBERS
2020-21
2 MATHEMA TICS
1.2 Euclid’s Division Lemma
Consider the following folk puzzle*.
A trader was moving along a road selling eggs. An idler who didn’t have
much work to do, started to get the trader into a wordy duel. This grew into a
fight, he pulled the basket with eggs and dashed it on the floor. The eggs broke.
The trader requested the Panchayat to ask the idler to pay for the broken eggs.
The Panchayat asked the trader how many eggs were broken. He gave the
following response:
If counted in pairs, one will remain;
If counted in threes, two will remain;
If counted in fours, three will remain;
If counted in fives, four will remain;
If counted in sixes, five will remain;
If counted in sevens, nothing will remain;
My basket cannot accomodate more than 150 eggs.
So, how many eggs were there? Let us try and solve the puzzle. Let the number
of eggs be a. Then working backwards, we see that a is less than or equal to 150:
If counted in sevens, nothing will remain, which translates to a = 7p + 0, for
some natural number p. If counted in sixes, a = 6q+ 5, for some natural number q.
If counted in fives, four will remain. It translates to a = 5w + 4, for some natural
number w.
If counted in fours, three will remain. It translates to a = 4s + 3, for some natural
number s.
If counted in threes, two will remain. It translates to a = 3t + 2, for some natural
number t.
If counted in pairs, one will remain. It translates to a = 2u + 1, for some natural
number u.
That is, in each case, we have a and a positive integer b (in our example,
b takes values 7, 6, 5, 4, 3 and 2, respectively) which divides a and leaves a remainder
r (in our case, r is 0, 5, 4, 3, 2 and 1, respectively), that is smaller than b. The
* This is modified form of a puzzle given in ‘Numeracy Counts!’ by A. Rampal, and others.
2020-21
REAL NUMBERS 3
moment we write down such equations we are using Euclid’s division lemma,
which is given in Theorem 1.1.
Getting back to our puzzle, do you have any idea how you will solve it? Yes! Y ou
must look for the multiples of 7 which satisfy all the conditions. By trial and error
(using the concept of LCM), you will find he had 119 eggs.
In order to get a feel for what Euclid’s division lemma is, consider the following
pairs of integers:
17, 6; 5, 12; 20, 4
Like we did in the example, we can write the following relations for each such
pair:
17 = 6 × 2 + 5 (6 goes into 17 twice and leaves a remainder 5)
5 = 12 × 0 + 5 (This relation holds since 12 is larger than 5)
20 = 4 × 5 + 0 (Here 4 goes into 20 five-times and leaves no remainder)
That is, for each pair of positive integers a and b, we have found whole numbers
q and r, satisfying the relation:
a = bq + r, 0 = r < b
Note that q or r can also be zero.
Why don’t you now try finding integers q and r for the following pairs of positive
integers a and b?
(i) 10, 3; (ii) 4, 19; (iii) 81, 3
Did you notice that q and r are unique? These are the only integers satisfying the
conditions a = bq + r, where 0 = r < b. You may have also realised that this is nothing
but a restatement of the long division process you have been doing all these years, and
that the integers q and r are called the quotient and remainder, respectively.
A formal statement of this result is as follows :
Theorem 1.1 (Euclid’s Division Lemma) : Given positive integers a and b,
there exist unique integers q and r satisfying a = bq + r, 0 = r < b.
This result was perhaps known for a long time, but was first recorded in Book VII
of Euclid’s Elements. Euclid’s division algorithm is based on this lemma.
2020-21
4 MATHEMA TICS
An algorithm is a series of well defined steps
which gives a procedure for solving a type of
problem.
The word algorithm comes from the name
of the 9th century Persian mathematician
al-Khwarizmi. In fact, even the word ‘algebra’
is derived from a book, he wrote, called Hisab
al-jabr w’al-muqabala.
A lemma is a proven statement used for
proving another statement.
Euclid’s division algorithm is a technique to compute the Highest Common Factor
(HCF) of two given positive integers. Recall that the HCF of two positive integers a
and b is the largest positive integer d that divides both a and b.
Let us see how the algorithm works, through an example first. Suppose we need
to find the HCF of the integers 455 and 42. We start with the larger integer, that is,
455. Then we use Euclid’s lemma to get
455 = 42 × 10 + 35
Now consider the divisor 42 and the remainder 35, and apply the division lemma
to get
42 = 35 × 1 + 7
Now consider the divisor 35 and the remainder 7, and apply the division lemma
to get
35 = 7 × 5 + 0
Notice that the remainder has become zero, and we cannot proceed any further.
We claim that the HCF of 455 and 42 is the divisor at this stage, i.e., 7. You can easily
verify this by listing all the factors of 455 and 42. Why does this method work? It
works because of the following result.
So, let us state Euclid’s division algorithm clearly.
To obtain the HCF of two positive integers, say c and d, with c > d, follow
the steps below:
Step 1 : Apply Euclid’s division lemma, to c and d. So, we find whole numbers, q and
r such that c = dq + r, 0 = r < d.
Step 2 : If r = 0, d is the HCF of c and d. If r ? 0, apply the division lemma to d and r.
Step 3 : Continue the process till the remainder is zero. The divisor at this stage will
be the required HCF.
Muhammad ibn Musa al-Khwarizmi
(C.E. 780 – 850)
2020-21
REAL NUMBERS 5
This algorithm works because HCF (c, d) = HCF (d, r) where the symbol
HCF (c, d) denotes the HCF of c and d, etc.
Example 1 : Use Euclid’s algorithm to find the HCF of 4052 and 12576.
Solution :
Step 1 : Since 12576 > 4052, we apply the division lemma to 12576 and 4052, to get
12576 = 4052 × 3 + 420
Step 2 : Since the remainder 420 ? 0, we apply the division lemma to 4052 and 420, to
get
4052 = 420 × 9 + 272
Step 3 : We consider the new divisor 420 and the new remainder 272, and apply the
division lemma to get
420 = 272 × 1 + 148
We consider the new divisor 272 and the new remainder 148, and apply the division
lemma to get
272 = 148 × 1 + 124
We consider the new divisor 148 and the new remainder 124, and apply the division
lemma to get
148 = 124 × 1 + 24
We consider the new divisor 124 and the new remainder 24, and apply the division
lemma to get
124 = 24 × 5 + 4
We consider the new divisor 24 and the new remainder 4, and apply the division
lemma to get
24 = 4 × 6 + 0
The remainder has now become zero, so our procedure stops. Since the divisor at this
stage is 4, the HCF of 12576 and 4052 is 4.
Notice that 4 = HCF (24, 4) = HCF (124, 24) = HCF (148, 124) =
HCF (272, 148) = HCF (420, 272) = HCF (4052, 420) = HCF (12576, 4052).
Euclid’s division algorithm is not only useful for calculating the HCF of very
large numbers, but also because it is one of the earliest examples of an algorithm that
a computer had been programmed to carry out.
Remarks :
1. Euclid’s division lemma and algorithm are so closely interlinked that people often
call former as the division algorithm also.
2. Although Euclid’s Division Algorithm is stated for only positive integers, it can be
extended for all integers except zero, i.e., b ? 0. However, we shall not discuss this
aspect here.
2020-21
```
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## Mathematics (Maths) Class 10

51 videos|346 docs|103 tests

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