These notes for CBSE Class 10 Mathematics, Chapter 1, focus on Real Numbers. They explain Euclid's Division Lemma, which states that for any two positive integers 'a' and 'b', there exist unique integers 'q' and 'r' such that a = bq + r, where 0 ? r < b. The notes also detail Euclid's Division Algorithm, a step-by-step method to find the Highest Common Factor (HCF) of two numbers. The Fundamental Theorem of Arithmetic is presented, stating that every composite number can be uniquely factorized into primes. The notes include examples and problems related to these concepts, such as finding HCF, expressing GCD as a linear combination, proving divisibility of consecutive integers, and demonstrating irrationality. These resources are designed to aid students in revising the key concepts of Real Numbers for their exams.
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1. Euclid’s Division lemma:- Given Positive integers a and b there exist unique integers q and r
satisfying
a=bq +r, where 0 where a, b, q and r are respectively called as dividend, divisor, quotient and
remainder.
2. Euclid’s division Algorithm:- To obtain the HCF of two positive integers say c and d, with c>0, follow
the steps below:
Step I: Apply Euclid’s division lemma, to c and d, so we find whole numbers, q and r such that c =dq+r, 0
Step II: If r=0, d is the HCF of c and d. If r
Step III: Continue the process till the remainder is zero. The divisor at this stage will be the required HC 3. The Fundamental theorem of Arithmetic:-
Every composite number can be expressed ( factorised ) as a product of primes, and this
factorization is unique, apart from the order in which the prime factors occur.
Ex.:
Theorem: LET be a rational number whose decimal expansion terminates. Then can be
expressed in the form
1. If the H C F of 657 and 963 is expressible in the form of 657x + 963x - 15 find x.Definition
(Ans:x=22)
2. Express the GCD of 48 and 18 as a linear combination. (Ans: Not unique) | Hence, x and y are not unique.
3. Prove that one of every three consecutive integers is divisible by 3.
Ans:
n,n+1,n+2 be three consecutive positive integers
We know that n is of the form 3q, 3q +1, 3q + 2
So we have the following cases Properties
4. Find the largest possible positive integer that will divide 398, 436, and 542 leaving
remainder 7, 11, 15 respectively.
(Ans: 17)
5. Find the least number that is divisible by all numbers between 1 and 10 (both
inclusive).
6. Show that 571 is a prime number.
7. If d is the HCF of 30, 72, find the value of x & y satisfying d = 30x + 72y.
Hence, x and y are not unique
8. Show that the product of 3 consecutive positive integers is divisible by 6.
Ans: Proceed as in question sum no. 3
9. Show that for odd positive integer to be a perfect square, it should be of the form
11. If a and b are positive integers. Show thatalways lies between
12. Prove thatis irrational, for every nN Symbol chart
Euclid's Division Lemma states that for any two positive integers 'a' (dividend) and 'b' (divisor), there exist unique integers 'q' (quotient) and 'r' (remainder) such that a = bq + r, where 0 ? r < b.
Euclid's Division Algorithm uses the lemma repeatedly. It applies the lemma to find the remainder, then uses the divisor as the new dividend and the remainder as the new divisor, continuing until the remainder is zero. The last non-zero divisor is the HCF.
The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a product of prime numbers, and this factorization is unique, except for the order of the prime factors.
Yes, the HCF of two numbers can be expressed in the form ax + by, where x and y are integers. However, these integers x and y are not always unique.
The notes suggest proving irrationality by assuming the number is rational and then deriving a contradiction, often involving properties of integers and prime factorization.
Yes, these are revision notes for CBSE Class 10 Mathematics, specifically for Chapter 1 on Real Numbers.
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