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chapter -1 : Real Numbers
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
chapter -1 : Real Numbers
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
