 # NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers  View Notes

## NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers - Free PDF

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Q1. In the equation a = bq + r of Euclid's division lemma what are the positive integers and unique terms? And Why are they called so?

According to Euclid’s division lemma, the equation a=bq+r consists of positive integers and unique terms. The terms a and b are positive integers. To satisfy the equation we need to find a value of q and r. Thus, q and r are two unique integers which satisfy the equation a=bq+r.

Example: Let a = 25 and b = 6. Find the value of q and r to satisfy the equation of Euclid’s division lemma.

Solution: 25=6xq+r.

To solve this equation, we need to divide 25 by 6. The resulting quotient and the remainder will be 1. Replacing the value in the equation: 25=6x4+1. Thus 4 and 1 are unique integers.

Q2: Vedantu explains why Euclid's division lemma is said to be a restatement of the long division process.

Euclid’s division is said to be a restatement of the long division process because to satisfy the equation of Euclid’s division lemma we need to divide ‘a’ by ‘b’ to get the value of q and r. On comparing we can say that the terms a, b, q and r are nothing dividend, divisor, quotient and remainder respectively.

Example: Find the integers q and r for the following pairs of positive pair of integers a and b.

i) 10, 3

Solution: 10 = 3xq+r

The resultant equation will be 10=3x3+1. Where q=3 and r=1

Thus, Euclid division lemma can be called as a restatement of the long division process.

What is the fundamental theorem of Arithmetics?

According to the Fundamental Theorem of Arithmetics, a composite number can be represented as a product of prime numbers, keeping the factorisation unique.

To understand this statement better let us find the factors of 630.

630 = 2x3x3x5x7.

Here, 630 is a composite number which can be represented as a product of 2, 3, 5 and 7. The factorisation of 630 into its prime numbers is unique. Fundamental Theorem of Arithmetic holds true for all the composite numbers. Vedantu explains all the theorems in Mathematics with examples and makes it easy for you to remember.

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Q4. Can a composite number can be represented as a product of prime numbers, keeping the factorisation unique? Explain with an example.

According to the Fundamental Theorem of Arithmetics, a composite number can be represented as a product of prime numbers, keeping the factorisation unique.

To understand this statement better let us take the factors of 630.

630 = 2x3x3x5x7.

Here, 630 is a composite number which can be represented as a product of 2, 3, 5 and 7. The factorisation of 630 into its prime numbers is unique. Fundamental Theorem of Arithmetic holds true for all the composite numbers.

Q5. What is Euclid’s division lemma and what is its use?

Euclid’s division lemma states that for given two positive integers a and b, there exist unique integers q and r which satisfy the equation: a = bq + r, where 0 ≤ r < b.

Example: let a = 26 and b = 5. Find the value of unique integers q and r which satisfy the Euclid’s division lemma.

According to Euclid’s division lemma, 26 = 5q + r

To solve the above equation, divide 26 by 5, the resulting quotient is 5 and remainder is 1. Putting these values together in the equation: 26 = 5(5) + 1. Thus, the value of unique integer q is 5 and that of r is 1.

Euclid’s division lemma is a proven statement and it is used for proving another statement. It is the basis for Euclid’s division algorithm, which is used to compute the Highest Common Factor (HCF) of two given positive integers.

Euclid’s division lemma/algorithm is also used for finding various properties of numbers.

Q6. What is fundamental theorem of arithmetics?

Fundamental theorem of arithmetics states that every composite number can be expressed or factored as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.

Example: let us find the factors of 210.

210 = 2 × 3 × 5 × 7

Here, 210 is a composite number, which is expressed as a product of prime numbers 2, 3, 5 and 7 and this factorisation is unique.

Q7. Without actually performing the long division, how can you check whether the given rational number has a terminating or a non-terminating repeating decimal expansion?

Let the given rational number be x = pq, then it has terminating or a non-terminating repeating decimal expansion can be checked by the prime factorisation of its denominator ‘q’.

So, the decimal expansion of rational number x = pwill terminate, if the prime factorisation of ‘q’ is of the form 2m5n,  where m, n are non-negative integers.

Otherwise, the decimal expansion of rational number x = pqwill  now-terminate or repeat, if the prime factorisation of ‘q’ is not of the form 2m5n,  where m, n are non-negative integers.

Example: The decimal expansion of 17343is non-terminating repeating because the prime factorisation of 343 = 7 × 7 × 7 = 73, which is not of the form  2m5n.

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