RD Sharma solutions for Mathematics [English] Class 10 chapter 1 - Real Numbers [Latest edition]

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.

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

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

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.

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

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.

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

Show that the square of an odd positive 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.

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

32 and 54

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

18 and 24

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

70 and 30

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

56 and 88

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

475 and 495

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

75 and 243

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

240 and 6552

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

155 and 1385

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

100 and 190

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

105 and 120

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

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

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

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

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

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

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

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

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

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?

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?

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?

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?

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

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

What is the largest number 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 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?

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 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?

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.

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

Express each of the following integers as a product of its prime factors:

420

Express each of the following integers as a product of its prime factors:

468

Express each of the following integers as a product of its prime factors:

945

Express each of the following integers as a product of its prime factors:

7325

Determine the prime factorisation of each of the following positive integer:

 20570

Determine the prime factorisation of each of the following positive integer:

58500

Determine the prime factorisation of each of the following positive integer:

45470971

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

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

Find the LCM and HCF of the following pair of integers and verify that LCM × HCF = Product of the two numbers.

26 and 91

Find the LCM and HCF of the following pair of integers and verify that LCM × HCF = product of the two numbers.

510 and 92

Find the LCM and HCF of the following pair of integers and verify that LCM × HCF = product of the two numbers.

336 and 54

Find the LCM and HCF of the following integers by applying the prime factorisation method.

12, 15 and 21

Find the LCM and HCF of the following integers by applying the prime factorisation method.

17, 23 and 29

Find the LCM and HCF of the following integers by applying the prime factorisation method.

8, 9 and 25

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

40, 36 and 126

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

 84, 90 and 120

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

 24, 15 and 36

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

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 least number that is divisible by all the numbers between 1 and 10 (both inclusive).

Show that the following numbers are irrational.

Show that the following numbers are irrational.

Show that the following numbers are irrational.

Show that the following numbers are irrational.

Prove that following numbers are irrationals:

Prove that following numbers are irrationals:

Prove that following numbers are irrationals:

Prove that following numbers are irrationals:

Show that \[2 - \sqrt{3}\] is an irrational number.

Show that \[3 + \sqrt{2}\] is an irrational number.

Prove that \[4 - 5\sqrt{2}\] is an irrational number.

Show that \[5 - 2\sqrt{3}\] is an irrational number.

Prove that \[2\sqrt{3} - 1\] is an irrational number.

Prove that \[2 - 3\sqrt{5}\] is an irrational number.

Prove that \[\sqrt{5} + \sqrt{3}\] is irrational.

Prove that for any prime positive integer p, \[\sqrt{p}\]

If p, q are prime positive integers, prove that \[\sqrt{p} + \sqrt{q}\] is an irrational number.

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

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

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

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

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

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

Write down the decimal expansions of the following rational numbers by writing their denominators in the form 2^m × 5ⁿ, where, m, n are non-negative integers. \[\frac{3}{8}\]

Write down the decimal expansions of the following rational numbers by writing their denominators in the form 2^m × 5ⁿ, where, m, n are non-negative integers.\[\frac{13}{125}\]

Write down the decimal expansions of the following rational numbers by writing their denominators in the form 2^m × 5ⁿ, where, m, n are non-negative integers.\[\frac{7}{80}\]

Write down the decimal expansions of the following rational numbers by writing their denominators in the form 2^m × 5ⁿ, where, m, n are non-negative integers.\[\frac{14588}{625}\]

Write down the decimal expansions of the following rational numbers by writing their denominators in the form 2^m × 5ⁿ, where, m, n are non-negative integers.\[\frac{129}{2^2 \times 5^7}\]

What can you say about the prime factorisations of the denominators of the following rationals: 43.123456789

What can you say about the prime factorisations of the denominators of the following rationals: \[43 . 123456789\]

​What can you say about the prime factorisations of the denominators of the following rationals: \[27 . \bar{{142857}}\]

​What can you say about the prime factorisations of the denominators of the following rationals: 0.120120012000120000 ...

Write the condition to be satisfied by q so that a rational number\[\frac{p}{q}\]has a terminating decimal expansions.

Write the condition to be satisfied by q so that a rational number\[\frac{p}{q}\]has a terminating decimal expansion.

Complete the missing entries in the following factor tree.

The decimal expansion of the rational number \[\frac{43}{2^4 \times 5^3}\] will terminate after how many places of decimals?

Has the rational number \[\frac{441}{2^2 \times 5^7 \times 7^2}\] a terminating or a nonterminating decimal representation?

Write whether \[\frac{2\sqrt{45} + 3\sqrt{20}}{2\sqrt{5}}\]

The exponent of 2 in the prime factorisation of 144, is

The LCM of two numbers is 1200. Which of the following cannot be their HCF?

If n = 2³ ✕ 3⁴ ✕ 5⁴ ✕ 7, then the number of consecutive zeros in n, where n is a natural number, is

The sum of the exponents of the prime factors in the prime factorisation of 196, is

The number of decimal place after which the decimal expansion of the rational number \[\frac{23}{2^2 \times 5}\] will terminate, is

If p₁ and p₂ are two odd prime numbers such that p₁ > p₂, then

If two positive ingeters a and b are expressible in the form a = pq² and b = p³q; p, q being prime number, then LCM (a, b) is

In Q.No. 7, HCF (a, b) is

If two positive integers m and n are expressible in the form m = pq³ and n = p³q², where p, q are prime numbers, then HCF (m, n) = 

If the LCM of a and 18 is 36 and the HCF of a and 18 is 2, then a = 

The HCF of 95 and 152, is

If HCF (26, 169) = 13, then LCM (26, 169) =

If a = 2³ ✕ 3, b = 2 ✕ 3 ✕ 5, c = 3ⁿ ✕ 5 and LCM (a, b, c) = 2³ ✕ 3² ✕ 5, then n =

The decimal expansion of the rational number \[\frac{14587}{1250}\]  will terminate after

If p and q are co-prime numbers, then p² and q² are

Which of the following rational numbers have terminating decimal?

If 3 is the least prime factor of number a and 7 is the least prime factor of number b, then the least prime factor of a + b, is ______.

The smallest number by which \[\sqrt{27}\]  should be multiplied so as to get a rational number is ______.

The smallest rational number by which \[\frac{1}{3}\] should be multiplied so that its decimal expansion terminates after one place of decimal, is

If n is a natural number, then 9²ⁿ − 4²ⁿ is always divisible by ______.

If n is any natural number, then 6ⁿ − 5ⁿ always ends with

The LCM and HCF of two rational numbers are equal, then the numbers must be

If the sum of LCM and HCF of two numbers is 1260 and their LCM is 900 more than their HCF, then the product of two numbers is

The remainder when the square of any prime number greater than 3 is divided by 6, is