Chapter 1 of CBSE Class 10 Maths, titled "Real Numbers," delves into the fundamental concepts of real numbers, including Euclid's Division Lemma and the Fundamental Theorem of Arithmetic. Students explore the properties of integers, rational numbers, and irrational numbers, and learn to express com...
S.No. Statement
ma Given positive integers a and b,
1. Let p be a prime number. Lem there exist unique integers q and r
on satisfying a = bq + r ; 0<r<b
If p divides a2, then p divides a, visi
i
where a is a positive integer Theorem
s D
Euclid’s
2. √2 is irrational Real
3. Let x be a rational number F D Steps to obtain the HCF of two
d
Numbers iv
un
positive integers, say c and d,
d
whose decimal expansion
tho
isi
with c>d
e
on
am
terminates. Then x can be Al
p go
en
nM
Step 1: Apply Euclid’s Division
t
expressed in the form q , rith
a
m Lemma, to c & d. c = dq + r
Arit
where p & q are coprime,
izatio
the prime factorisation of q Step 2: If r = zero, d is the HCF of
l The
hmet
is of the form 2n, 5m where n, m c and d
ic
are non-negative integers If r ≠ 0, apply Euclid’s
Division to d and r
p
orem of
4. Let x = q be a rational number
Step 3: Continue the process till
Oswaal NCERT Exemplar Problems–Solutions, MATHEMATICS, Class-X
Prime Factor
such that the prime factorisation the remainder is zero
of q is of the form 2n, 5m where Every composite number can be
n, m are non-negative integers. expressed as a product of primes,
For any two positive
Then, x has a decimal expansion and this factorisation is unique,
integers, a and b
which terminates. apart from the order in which the
HCF (a, b) × LCM (a, b)
prime factors occur
p =a×b
5. Let x = q be a rational number, For Example
Composite Number x = P1P2 ... Pn,
such that the prime factorisation f(x) = 3x2y
where P1P2 ... Pn are prime numbers
g(x) = 6xy2
of q is not of the form of 2n5m HCF = 3xy
where n, m are non-negative LCM = 6x2y2
integers. Then, x has a decimal
expansion which is
non-terminating repeating
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