Solution of Chapter 1. Real Numbers (RD Sharma - Mathematics Book)

If a and b are two odd positive integers such that a > b, then prove that one of the two numbers and is odd and the other is even.

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

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

For any positive integer, prove that 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 HCF of two positive integers and find the HCF of the following pairs of numbers:

(i) 32 and 54 (ii) 18 and 24

(iii) 70 and 30 (iv) 56 and 88

(v) 475 and 495 (vi) 75 and 243

(vii)240 and 6552(viii)155 and 1385

(ix) 100 and 190 (x) 105 and 120

Use Euclid’s division algorithm to find the HCF of

(i) 135 and 225 (ii)196 & 38220

(iii) 867 & 255.

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

(i) 963 & 657 (ii) 592 & 252

(iii) 506 & 1155 (iv) 1288 & 575

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

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 657 x + 963 × -15, find x.

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

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 remainder 4, 5 and 6 respectively.

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

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 litres of oil of one kind, 180 litres of another kind and 240 litres 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?

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

(i) 420 (ii) 468

(iii) 945 (iv) 7325

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

(i) 20570 (ii) 58500

(iii) 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 numbers n.

Find the LCM and HCF of the following pairs of integers and verify that LMC × 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 12

(vi) 24, 15 and 36

Given that HCF (306, 657) = 9, find LCM (360, 657).

Can two numbers have 16 as their HCF and 380 as their LCM? Give reason.

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.

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

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

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

Find the smallest number which leaves remainders 8 and 12 when divided by 28 and 32 respectively.

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

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

A rectangular courtyard is 18 m 72 cm long and 13 m 20 cm broad. It is to be paved with square tiles of the same size. Find the least possible number of such tiles.

A circular field has a circumference of 360 km. Three cyclists start together and can cycle 48, 60 and 72 km a day, round the field. When will they meet again?

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) (ii)

(iii) 6 + (iv) 3 -

Prove that following numbers are irrationals:

(i) (ii)

(iii) (iv)

Show that is an irrational numbers.

Show that is an irrational number.

Prove that is an irrational number.

Show that is an irrational number.

Prove that is an irrational number.

Prove that is an irrational number.

Prove that is irrational.

Prove that is an irrational number.

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

If p, q are prime positive integers, prove that 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

(i) (ii)

(iii) (iv)

(v)

Write down the decimal expansions of the following rational numbers by writing their denominators in the form, where m, n are non-negative integers.

(i) (ii)

(iii) (iv)

(v)

What can you say about the prime factorizations of the denominators of the following rational?

(i) 43.123456789

(ii)

(iii)

(iv) 0.120120012000120000…

State Euclid’s division lemma.

State Fundamental Theorem of Arithmetic.

Write 98 as product of its prime factors.

Write the exponent of 2 in the prime factorization of 144.

Write the sum of the exponents of prime factors in the prime factorization of 98.

If the prime factorization of a natural number n is 2³ × 3² × 5² × 7, write the number of consecutive zeros in n.

If the product of two numbers is 1080 and their HCF is 30, find their LCM.

Write the condition to be satisfied by q so that a rational number has a terminating 9 decimal expansion.

Write the condition to be satisfied by q so that a rational number has a non-terminating terminating decimal expansion.

Complete the missing entries in the following factor tree.

The decimal expansion of the rational number will terminate after how many places of decimals?

Has the rational number a terminating or a non terminating decimal representation?

Write whether on simplification gives a rational or an irrational number.

What is an algorithm?

What is a lemma?

If p and q are two prime numbers, then what is their HCF?

If p and q are two prime numbers, then what is their LCM?

What is the total number of factors of a prime number?

What is a composite number?

What is the HCF of the smallest composite number and the smallest prime number?

HCF of two numbers is always a factor of their LCM (True/False).

π is an irrational number (True/False).

The sum of two prime numbers is always a prime number (True/False).

The product of any three consecutive natural numbers is divisible by 6 (True/False).

Every even integer is of the form 2m, where m is an integer (True/False).

Every odd integer is of the form 2m - 1, where m is an integer (True/False).

The product of two irrational numbers is an irrational number (True/False).

The sum of two irrational numbers is an irrational number (True/False).

For what value of n, 2ⁿ x 5ⁿ ends in 5.

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

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

Two numbers have 12 as their HCF and 350 as their LCM (True/False).

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 places after which the decimal expansion of the rational number will terminate, is

If p₁ and p₂ are two odd prime numbers such that p₁ > p₂, then p₁² – p₂² is

If two positive integers a and b are expressible in the form a = pq² and b = p³ q; p, q being prime numbers, 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 = p³ q² 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 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?

(i)16/225 (ii)5/18

(iii)2/21 (iv)7/250

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

is

The smallest number by which should be multiplied so as to get a rational number is

The smallest rational number by which 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