Revision Notes
Class - 10 Maths
Chapter 1 - Real Numbers
● Real numbers:
- All rational and irrational numbers taken together make the real
numbers. On the number line, any real number can be plotted.
● Euclid’s Division Lemma:
- A lemma is a verified statement that is utilised to prove another.
Euclid’s Division Lemma states that for any two integers a and b ,
there exists a unique pair of integers q and r such that a = b × q + r
where 0r < b.
- The lemma can be simply stated as :
Dividend = Divisor × Quotient + Remainder
- For any pair of dividend and divisor, the quotient and remainder
obtained are going to be unique.
● Euclid’s Division Algorithm:
- An algorithm is a set of well-defined steps that describe how to solve a
certain problem. The Highest Common Factor (HCF) of two positive
integers is computed using Euclid's division algorithm.
- Follow the steps below to find the HCF of two positive integers, say c
and d , with c > d:
Step 1: We apply Euclid’s Division Lemma to find two integers q
and r such that c = d × q + r where 0r <d .
Step 2: If r = 0, the H.C.F is d , else, we apply Euclid’s division
Lemma to d (the divisor) and r (the remainder) to get another pair of
quotient and remainder.
Step 3: Repeat Steps 1–3 until the remainder is zero. The needed HCF
will be the divisor at the last step.
● The Fundamental Theorem of Arithmetic:
The process of expressing a natural number as a product of prime
numbers is known as prime factorization.
Apart from the sequence in which the prime components occur, the
prime factorisation for a given number is unique.
Class X Maths www.vedantu.com 1
Class - 10 Maths
Chapter 1 - Real Numbers
● Real numbers:
- All rational and irrational numbers taken together make the real
numbers. On the number line, any real number can be plotted.
● Euclid’s Division Lemma:
- A lemma is a verified statement that is utilised to prove another.
Euclid’s Division Lemma states that for any two integers a and b ,
there exists a unique pair of integers q and r such that a = b × q + r
where 0r < b.
- The lemma can be simply stated as :
Dividend = Divisor × Quotient + Remainder
- For any pair of dividend and divisor, the quotient and remainder
obtained are going to be unique.
● Euclid’s Division Algorithm:
- An algorithm is a set of well-defined steps that describe how to solve a
certain problem. The Highest Common Factor (HCF) of two positive
integers is computed using Euclid's division algorithm.
- Follow the steps below to find the HCF of two positive integers, say c
and d , with c > d:
Step 1: We apply Euclid’s Division Lemma to find two integers q
and r such that c = d × q + r where 0r <d .
Step 2: If r = 0, the H.C.F is d , else, we apply Euclid’s division
Lemma to d (the divisor) and r (the remainder) to get another pair of
quotient and remainder.
Step 3: Repeat Steps 1–3 until the remainder is zero. The needed HCF
will be the divisor at the last step.
● The Fundamental Theorem of Arithmetic:
The process of expressing a natural number as a product of prime
numbers is known as prime factorization.
Apart from the sequence in which the prime components occur, the
prime factorisation for a given number is unique.
Class X Maths www.vedantu.com 1
