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# RD Sharma solutions for Class 10th Board Exam Mathematics chapter 1 - Real Numbers

## Chapter 1 : Real Numbers

Ex. 1.1Ex. 1.2

#### Page 10

Ex. 1.1 | Q 1 | Page 10

If a and b are two odd positive integers such that a > b, then prove that one of the two numbers (a+b)/2and(a-b)/2 is odd and the other is even.

Ex. 1.1 | Q 2 | Page 10

Prove that the product of two consecutive positive integers is divisible by 2.

Ex. 1.1 | Q 3 | Page 10

Prove that the product of three consecutive positive integer is divisible by 6.

Ex. 1.1 | Q 4 | Page 10

For any positive integer n , prove that n3 − n divisible by 6.

Ex. 1.1 | Q 5 | Page 10

Prove that if a positive integer is of the form 6q + 5, then it is of the form 3q + 2 for some integer q, but not conversely.

Ex. 1.1 | Q 6 | Page 10

Prove that the square of any positive integer of the form 5q + 1 is of the same form.

Ex. 1.1 | Q 7 | Page 10

Prove that the square of any positive integer is of the form 3m or, 3m + 1 but not of the form 3m +2.

Ex. 1.1 | Q 8 | Page 10

Prove that the square of any positive integer is of the form 4q or 4q + 1 for some integer q.

Ex. 1.1 | Q 9 | Page 10

Prove that the square of any positive integer is of the form 5q, 5q + 1, 5q + 4 for some integer q.

Ex. 1.1 | Q 10 | Page 10

Show that the square of an odd positive integer is of the form 8q + 1, for some integer q.

Ex. 1.1 | Q 11 | Page 10

Show that any positive odd integer is of the form 6q + 1 or, 6q + 3 or, 6q + 5, where q is some integer.

#### Pages 27 - 28

Ex. 1.2 | Q 1.01 | Page 27

Define HOE of two positive integers and find the HCF of the following pair of numbers:

32 and 54

Ex. 1.2 | Q 1.02 | Page 27

Define HOE of two positive integers and find the HCF of the following pair of numbers:

18 and 24

Ex. 1.2 | Q 1.03 | Page 27

Define HOE of two positive integers and find the HCF of the following pair of numbers:

70 and 30

Ex. 1.2 | Q 1.04 | Page 27

Define HOE of two positive integers and find the HCF of the following pair of numbers:

56 and 88

Ex. 1.2 | Q 1.05 | Page 27

Define HOE of two positive integers and find the HCF of the following pair of numbers:

475 and 495

Ex. 1.2 | Q 1.06 | Page 27

Define HOE of two positive integers and find the HCF of the following pair of numbers:

75 and 243

Ex. 1.2 | Q 1.07 | Page 27

Define HOE of two positive integers and find the HCF of the following pair of numbers:

240 and 6552

Ex. 1.2 | Q 1.08 | Page 27

Define HOE of two positive integers and find the HCF of the following pair of numbers:

155 and 1385

Ex. 1.2 | Q 1.09 | Page 27

Define HOE of two positive integers and find the HCF of the following pair of numbers:

100 and 190

Ex. 1.2 | Q 1.1 | Page 27

Define HOE of two positive integers and find the HCF of the following pair of numbers:

105 and 120

Ex. 1.2 | Q 2.1 | Page 27

Using Euclid's division algorithm, find the H.C.F. of 135 and 225

Ex. 1.2 | Q 2.2 | Page 27

Using Euclid's division algorithm, find the H.C.F. of 196 and 38220

Ex. 1.2 | Q 3.1 | Page 27

Find the HCF of the following pairs of integers and express it as a linear combination of 963 and 657.

Ex. 1.2 | Q 3.2 | Page 27

Find the HCF of the following pairs of integers and express it as a linear combination of 592 and 252.

Ex. 1.2 | Q 3.3 | Page 27

Find the HCF of the following pairs of integers and express it as a linear combination of 506 and 1155.

Ex. 1.2 | Q 3.4 | Page 27

Find the HCF of the following pairs of integers and express it as a linear combination of 1288 and 575.

Ex. 1.2 | Q 4 | Page 27

Find the largest number which divides 615 and 963 leaving remainder 6 in each case.

Ex. 1.2 | Q 5 | Page 27

If the HCF of 408 and 1032 is expressible in the form 1032 m − 408 × 5, find m.

Ex. 1.2 | Q 6 | Page 27

If the HCF of 657 and 963 is expressible in the form 657x + 963y − 15, find x.

Ex. 1.2 | Q 7 | Page 27

An army contingent of 616 members is to march behind an army band of 32 members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march?

Ex. 1.2 | Q 8 | Page 27

A merchant has 120 liters of oil of one kind, 180 liters of another kind and 240 liters of third kind. He wants to sell the oil by filling the three kinds of oil in tins of equal capacity. What should be the greatest capacity of such a tin?

Ex. 1.2 | Q 9 | Page 27

During a sale, colour pencils were being sold in packs of 24 each and crayons in packs of 32 each. If you want full packs of both and the same number of pencils and crayons, how many of each would you need to buy?

Ex. 1.2 | Q 10 | Page 28

144 cartons of Coke Cans and 90 cartons of Pepsi Cans are to be stacked in a Canteen. If each stack is of the same height and is to contain cartons of the same drink, what would be the greatest number of cartons each stack would have?

Ex. 1.2 | Q 11 | Page 28

Find the greatest number which divides 285 and 1249 leaving remainders 9 and 7 respectively.

Ex. 1.2 | Q 12 | Page 28

Find the largest number which exactly divides 280 and 1245 leaving remainders 4 and 3, respectively.

Ex. 1.2 | Q 13 | Page 28

What is the largest number that divides 626, 3127 and 15628 and leaves remainders of 1, 2 and 3 respectively.

Ex. 1.2 | Q 14 | Page 28

Find the greatest number that will divide 445, 572 and 699 leaving remainders 4, 5 and 6 respectively.

Ex. 1.2 | Q 15 | Page 28

Find the greatest number which divides 2011 and 2623 leaving remainders 9 and 5 respectively.

Ex. 1.2 | Q 17 | Page 28

Two brands of chocolates are available in packs of 24 and 15 respectively. If I need to buy an equal number of chocolates of both kinds, what is the least number of boxes of each kind I would need to buy?

Ex. 1.2 | Q 18 | Page 28

A mason has to fit a bathroom with square marble tiles of the largest possible size. The size of the bathroom is 10 ft. by 8 ft. What would be the size in inches of the tile required that has to be cut and how many such tiles are required?

Ex. 1.2 | Q 19 | Page 28

15 pastries and 12 biscuit packets have been donated for a school fete. These are to be packed in several smaller identical boxes with the same number of pastries and biscuit packets in each. How many biscuit packets and how many pastries will each box contain?

Ex. 1.2 | Q 20 | Page 28

105 goats, 140 donkeys and 175 cows have to be taken across a river. There is only one boat which will have to make many trips in order to do so. The lazy boatman has his own conditions for transporting them. He insists that he will take the same number of animals in every trip and they have to be of the same kind. He will naturally like to take the largest possible number each time. Can you tell how many animals went in each trip?

Ex. 1.2 | Q 21 | Page 28

The length, breadth and height of a room are 8 m 25 cm, 6 m 75 cm and 4 m 50 cm, respectively. Determine the longest rod which can measure the three dimensions of the room exactly.

Ex. 1.2 | Q 22 | Page 28

Express the HCF of 468 and 222 as 468x + 222y where x, y are integers in two different ways.

## RD Sharma solutions for Class 10th Board Exam Mathematics chapter 1 - Real Numbers

RD Sharma solutions for Class 10th Board Exam 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 10 Mathematics solutions in a manner that help students grasp basic concepts better and faster.

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

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