R d sharma solutions mathematics class 10 Solutions for Chapter 1 real numbers

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

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

For any positive integer n, prove that \mathbf{n}^{3}-\mathbf{n} \text { divisible by } 6

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

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

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

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

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

Show that the square of any positive integer cannot be of the form 6m + 2 or 6m + 5 for any integer m.

Show that the cube of a positive integer of the form 6q + r, q is an integer and r = 0, 1, 2, 3, 4, 5 is also of the form 6m + r.

Show that the square of an odd integer can be of the form 6q + 1 or 6q + 3, for some integer q.

Show that the square of any positive integer cannot be of the form 3m + 2, where m is a natural number.

Use Euclid’s division algorithm to find the HCF of

(i) 135 and 225

(ii) 196 and 38220

(iii) 867 and 255

(iv) 184, 230 and 276

(v) 136, 170 and 255

Find the HCF of the following pair of integers and express it as a linear combination of them,

(i) 963 and 657

(ii) 592 and 252

(iii) 506 and 1155

(iv) 1288 and 575

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

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

During a sale, color 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?

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

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

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

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?

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

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?

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 on 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 me how many animals went on each trip?

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

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.

1. Express each of the following integers as a product of its prime.

(i) 420

(ii) 468

(iii) 945

(iv) 7325

Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = Product of the integers:

(i) 26 and 91

(ii) 510 and 92

(iii) 336 and 54

Find the LCM and HCF of the following integers by applying the prime factorization method:

(i) 12, 15 and 21

(ii) 17, 23 and 29

(iii) 8, 9 and 25

iv) 40, 36 and 126

(v) 84, 90 and 120

Given that HCF (306, 657) = 9 , find LCM ( 306, 657 )

Explain why 7 x 11 x 13 + 13 and 7 x 6 x 5 x 4 x 3 x 2 x 1 + 5 are composite numbers

Check whether 6n can end with the digit 0 for any natural number n.

Explain why 3 × 5 × 7 + 7 is a composite number.

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

The HCF of two numbers is 16 and their product is 3072. Find their LCM.

The LCM and HCF of two numbers are 180 and 6 respectively. If one of the numbers is 30, find the other number

The LCM and HCF of two numbers are 180 and 6 respectively. If one of the numbers is 30, find the other number

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

What is the smallest number that, when divided by 35, 56 and 91 leaves remainders of 7 in each case?

Find the greatest number of 6 digits exactly divisible by 24, 15, and 36.

Determine the number nearest to 110000 but greater 100000 which is exactly divisible by each of 8, 15, and 21.

In a morning walk, three persons step off together, their steps measure 80 cm, 85 cm, and 90 cm respectively. What is the minimum distance each should walk so that he can cover the distance in complete steps?

Show that the following numbers are irrational

(i) 1/√2

(ii) 7√5

(iii) 6 + √2

(iv) 3 − √5

Show that 2 − √3 is an irrational number

Show that 3 + √2 is an irrational number

Prove that 4 − 5√2 is an irrational number.

Show that 5 − 2√3 is an irrational number

Prove that 2√3 − 1 is an irrational number

Prove that 2 − 3√5 is an irrational number

Prove that √2 + √3 is irrational

Prove that for any prime positive integer p, √p is an irrational number

If p, q are prime positive integers, prove that √p + √q is an irrational number

Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion.

(i)23/8

ii) 125/441

(iii) 35/50
(iv) 77/210

(v) 129/(22 x 57 x 717)

Write the denominator of the rational number 257/5000 in the form 2^{m } \times 5^{\mathrm{n}}where m, n are non-negative integers. Hence, write the decimal expansion, without actual division

What can you say about the prime factorization of the denominators of the following rational:

(i) 43.123456789

(II)

(iv) 0.120120012000120000....