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# R.S. Aggarwal solutions for Class 10 Mathematics chapter 1 - Real Numbers

## Chapter 1: Real Numbers

#### Chapter 1: Real Numbers solutions [Page 0]

What do you mean by Euclid’s division algorithm?

A number when divided by 61 gives 27 as quotient and 32 as remainder. Find the number.

By what number should be 1365 be divided to get 31 as quotient and 32 as remainder?

Using Euclid’s algorithm, find the HCF of 405 and 2520 .

Using Euclid’s algorithm, find the HCF of  504 and 1188 .

Using Euclid’s algorithm, find the HCF of  960 and 1575 .

Show that every positive integer is either even or odd?

Show that every positive even integer is of the form (6m+1) or (6m+3) or (6m+5)where m is some integer.

Show that every positive even integer is of the form 4m and that every positive odd integer is of the form 4m + 1 for some integer m.

#### Chapter 1: Real Numbers solutions [Page 0]

Using prime factorization, find the HCF and LCM of  36, 84 In case verify that HCF × LCM = product of given numbers.

Using prime factorization, find the HCF and LCM of 23, 31 In case verify that HCF × LCM = product of given numbers.

Using prime factorization, find the HCF and LCM of  96, 404  In case verify that HCF × LCM = product of given numbers.

Using prime factorization, find the HCF and LCM of  144, 198  In case verify that HCF × LCM = product of given numbers.

Using prime factorization, find the HCF and LCM of  396, 1080  In case verify that HCF × LCM = product of given numbers.

Using prime factorization, find the HCF and LCM of  1152, 1664 In case verify that HCF × LCM = product of given numbers.

Using prime factorization, find the HCF and LCM of  8, 9, 25 .

Using prime factorization, find the HCF and LCM of  12,15, 21 .

Using prime factorization, find the HCF and LCM of  17,23,29 .

Using prime factorization, find the HCF and LCM of  24, 36, 40 .

Using prime factorization, find the HCF and LCM of  30, 72, 432 .

Using prime factorization, find the HCF and LCM of  21, 28, 36, 45 .

The HCF of two numbers is 23 and their LCM is 1449. If one of the numbers is 161, find the other.

The HCF of two numbers is 145 and their LCM is 2175. If one of the numbers is 725, find
the other.

The HCF of two numbers is 18 and their product is 12960. Find their LCM.

Is it possible to have two numbers whose HCF is 18 and LCM is 760?
Give reason.

Find the simplest form of 69 /92 .

Find the simplest form of 473/645 .

Find the simplest form of 1095 / 1168 .

Find the simplest form of  368 /496 .

Find the largest number which divides 438 and 606 leaving remainder 6 in each case.

Find the largest number which divides 320 and 457 leaving remainders 5 and 7 respectively.

Find the least number which when divides 35, 56 and 91 leaves the same remainder 7 in each case.

Find the smallest number which when divides 28 and 32, leaving remainders 8 and 12 respectively.

Find the smallest number which when increased by 17 is exactly divisible by both 468 and 520

Find the greatest number of four digits which is exactly divisible by 15, 24 and 36.

In a seminar, the number of participants in Hindi, English and mathematics are 60, 84 and 108 respectively. Find the minimum number of rooms required, if in each room, the same number of participants are to be seated and all of them being in the same subject .

Three sets of English, Mathematics and Science books containing 336, 240 and 96 books respectively have to be stacked in such a way that all the books are stored subject wise and the height of each stack is the same. How many stacks will be there?

Three pieces of timber 42m, 49m and 63m long have to be divided into planks of the same length. What is the greatest possible length of each plank? How many planks are formed?

Find the greatest possible length which can be used to measure exactly the lengths 7m, 3m 85cm and 12m 95cm

Find the maximum number of students among whom 1001 pens and 910 pencils can be distributed in such a way that each student gets the same number of pens and the same number of pencils.

Find the least number of square tiles required to pave the ceiling of a room 15m 17cm long and 9m 2cm broad.

Three measuring rods are 64 cm, 80 cm and 96 cm in length. Find the least length of cloth that can be measured an exact number of times, using any of the rods.

An electronic device makes a beep after every 60 seconds. Another device makes a beep after every 62 seconds. They beeped together at 10 a.m. At what time will they beep together at the earliest?

Six bells commence tolling together and toll at intervals of 2, 4, 6, 8, 10, 12 minutes respectively. In 30 hours, how many times do they toll together?

Find the missing numbers in the following factorization:

#### Chapter 1: Real Numbers solutions [Page 0]

Without actual division, show that each of the following rational numbers is a terminating decimal. Express each in decimal form.

(i) 23/(2^3 × 5^2)

Without actual division, show that each of the following rational numbers is a terminating decimal. Express each in decimal form.

(i) 24/125

Without actual division, show that each of the following rational numbers is a terminating decimal. Express each in decimal form.

(i) 171/800

Without actual division, show that each of the following rational numbers is a terminating decimal. Express each in decimal form.

(i) 151/600

Without actual division, show that each of the following rational numbers is a terminating decimal. Express each in decimal form.

(i) 17 /320

Without actual division, show that each of the following rational numbers is a terminating decimal. Express each in decimal form.

(i) 19/3125

Without actual division show that each of the following rational numbers is a non-terminating repeating decimal.

(i) 11/(2^3× 3)

Without actual division show that each of the following rational numbers is a non-terminating repeating decimal

(i) 73/(2^3× 3^3 × 5)

Without actual division show that each of the following rational numbers is a non-terminating repeating decimal.

(i) 129/(2^2× 5^7 × 7^5)

Without actual division show that each of the following rational numbers is a non-terminating repeating decimal.

(i) 9/35

Without actual division show that each of the following rational numbers is a non-terminating repeating decimal.

(i)77/210

Without actual division show that each of the following rational numbers is a non-terminating repeating decimal.

(i) 32/147

Without actual division show that each of the following rational numbers is a non-terminating repeating decimal.

(i) 29/343

Without actual division show that each of the following rational numbers is a non-terminating repeating decimal.

(i)64/455

Express each of the following as a rational number in its simplest form:

(i) 0.bar (8)

Express each of the following as a rational number in its simplest form:
(i) 2. bar(4)

Express each of the following as a rational number in its simplest form:

(i) 0. bar (24)

Express each of the following as a rational number in its simplest form:

(i) 0. bar (12)

Express each of the following as a rational number in its simplest form:

(i) 2. bar (24)

Express each of the following as a rational number in its simplest form:

(i)  0. bar(365)

#### Chapter 1: Real Numbers solutions [Page 0]

Define  rational numbers .

Define irrational numbers .

Define real numbers .

Classify the numbers  22/7 as rational or irrational:

Classify the numbers  3.1416 as rational  or irrational :

Classify the numbers  π as   rational or irrational:

Classify the numbers 3. bar(142857) as rational or irrational  :

Classify the numbers  5.636363 as rational or irrational   :

Classify the  numbers  2.040040004 as rational or irrational:

Classify the numbers 1.535335333  as rational or irrational:

Classify the numbers  3.121221222 as rational or irrational:

Classify the numbers  sqrt (21) as rational or irrational:

Classify the numbers root (3)(3) as rational or irrational:

Prove that of the numbers sqrt (6)  is irrational:

Prove that of the numbers   2 - sqrt(3)  is irrational:

Prove that of the numbers  3 + sqrt (2)  is irrational:

Prove that of the numbers  2 + sqrt (5) is irrational:

Prove that of the numbers 5 + 3 sqrt (2)  is irrational:

Prove that of the numbers  3 sqrt(7)  is irrational:

Prove that of the numbers 3/sqrt(5) is irrational:

Prove that of the numbers 2 -3 sqrt(5) is irrational:

Prove that of the numbers sqrt(3) + sqrt(5) is irrational:

Prove that 1/sqrt (3) is irrational.

Give an example of two irrationals whose sum is rational.

Give an example of two irrationals whose product is rational.

State whether the given statement is true or false:

(1)   The sum of two rationals is always rational

State whether the given statement is true or false:

1 . The product of two rationals is always rational

State whether the given statement is true or false:

1 .The sum of two irrationals is an irrational

State whether the given statement is true or false:

1 .The product of two irrationals is an irrational .

State whether the given statement is true or false:

1 . The sum of a rational and an irrational is irrational .

State whether the given statement is true or false:

1 . The product of a rational and an irrational is irrational .

Prove that (2 sqrt(3) – 1) is irrational.

Prove that (4 - 5sqrt(2) ) is irrational.

Show that (5 - 2sqrt(3)) is irrational.

Prove that 5sqrt(2) is irrational.

Show that 2sqrt(7) is irrational.

#### Chapter 1: Real Numbers solutions [Page 0]

What do you mean by Euclid’s division algorithm.

State fundamental theorem of arithmetic?

Express 360 as product of its prime factors

If a and b are two prime numbers then find the HCF(a, b)

If a and b are two prime numbers then find the HCF(a, b)

The product of two numbers is 1050 and their HCF is 25. Find their LCM.

What is a composite number?

If a and b are relatively prime then what is their HCF?

If the rational number a/bhas a terminating decimal expansion, what is the condition to be satisfied by b?

Find the simplest form of (2sqrt(45)+3sqrt(20))/(2sqrt(5))

Write the decimal expansion of 73/ ((2^4×5^3))

Show that there is no value of n for which (2^n xx  5^n)  ends in 5.

Is it possible to have two numbers whose HCF if 25 and LCM is 520?

Give an example of two irrationals whose sum is rational.

Give an example of two irrationals whose product is rational.

If a and b are relatively prime, what is their LCM?

The LCM of two numbers is 1200, show that the HCF of these numbers cannot be 500. Why ?

Express 0.bar(4) as a rational number simplest form.

Express 0.bar (23) as a rational number in simplest form.

Explain why 0.15015001500015……. is an irrational form.

Show that sqrt (2)/3 is irrational.

Write a rational number betweensqrt(3) and 2

Explain why 3. sqrt(1416) is a rational number ?

## R.S. Aggarwal solutions for Class 10 Mathematics chapter 1 - Real Numbers

R.S. Aggarwal solutions for Class 10 Maths chapter 1 (Real Numbers) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Secondary School Mathematics for Class 10 (for 2019 Examination) solutions in a manner that help students grasp basic concepts better and faster.

Further, we at Shaalaa.com provide such solutions so that students can prepare for written exams. R.S. Aggarwal textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

Concepts covered in Class 10 Mathematics chapter 1 Real Numbers are Introduction of Real Numbers, Real Numbers Examples and Solutions, Euclid’s Division Lemma, Fundamental Theorem of Arithmetic, Fundamental Theorem of Arithmetic Motivating Through Examples, Proofs of Irrationality, Revisiting Irrational Numbers, Revisiting Rational Numbers and Their Decimal Expansions.

Using R.S. Aggarwal Class 10 solutions Real Numbers exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in R.S. Aggarwal Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 10 prefer R.S. Aggarwal Textbook Solutions to score more in exam.

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