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

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.

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 one and only one of n, n + 1 and n + 2 is divisible by 3, where n is a positive integer.

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

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

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

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 is of the form 6q + r, where q is an integer and r = 0, 1, 2, 3, 4, 5.

Show that one and only one out of n, n + 4, n + 8, n + 12 and n + 16 is divisible by 5, where n is any positive integer.

[Hint: Any positive integer can be written in the form 5q, 5q + 1, 5q + 2, 5q + 3, 5q + 4].

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

A positive integer is of the form 3q + 1, q being a natural number. Can you write its square in any form other than 3m + 1, i.e., 3m or 3m + 2 for some integer m? Justify your answer.

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

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.

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

475 and 495

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

75 and 243

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

240 and 6552

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

155 and 1385

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

100 and 190

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

105 and 120

Using Euclid's division algorithm to find the HCF of 135 and 225.   

Use Euclid's division algorithm to find the HCF of 867 and 255.

Using Euclid’s division algorithm to find the HCF of 1260 and 7344.

Use Euclid's division algorithm to find the HCF of 2048 and 960.

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.

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

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?

Use Euclid's division algorithm to find the HCF of 184, 230 and 276.

Use Euclid's division algorithm to find the HCF of 136, 170 and 255.

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.

Using Euclid’s division algorithm, find the largest number that divides 1251, 9377 and 15628 leaving remainders 1, 2 and 3, respectively.

A merchant has 120 litres of oil of one kind, 180 litres of another kind and 240 litres of a 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?

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

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

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

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

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

420

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

468

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

945

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

7325

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

`1/sqrt(5)`

Determine the prime factorisation of the following positive integer:

 20570

Determine the prime factorisation of the following positive integer:

58500

Determine the prime factorisation of the following positive integer:

45470971

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

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

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

Can the number (15)ⁿ, n being a natural number, end with the digit 0? Give reasons.

Find the values of x, y, z and w in the following factor tree. Also write the prime factorisation of x.

State true or false for the following statement and justify your answer:

2 × 3 × 5 × 7 + 7 is a composite number

State true or false for the following statement and justify your answer:

2 × 3 × 5 × 7 + 1 is a composite number

Let p, q and r be three distinct prime numbers.

Check whether p × q × r + q is a composite number or not.

Further, give an example for 3 distinct primes p, q, r such that

1. p × q × r + 1 is a composite number.
2. p × q × r + 1 is a prime number.

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

336 and 54

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

404 and 96

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

96 and 120

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

72 and 120

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:

26, 65 and 117

Find by prime factorisation the LCM of the numbers 18180 and 7575. Also, find the HCF of the two numbers.

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

Write the smallest number which is divisible by both 306 and 657.

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

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

Three bells ring at intervals of 6, 12 and 18 minutes. If all the three bells rang at 6 AM, when will they ring together again?

Find the largest number which on dividing 1251, 9377 and 15628 leave remainders 1, 2 and 3 respectively.

On a morning walk, three persons step out together and their steps measure 30 cm, 36 cm, and 40 cm respectively. What is the minimum distance each should walk so that each can cover the same distance in complete steps?

The traffic lights at three different road crossings change after every 48 seconds, 72 seconds and 108 seconds. If they change simultaneously at 7 a.m., at what time will they change together next?

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.

Show that the following numbers are irrational.

`1/sqrt(5)`

Prove that following number is irrational:

Prove that following number is irrational:

Prove that following number is irrational:

Prove that following number is irrational:

Show that `3 + sqrt(2)` is an irrational number.

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

Prove that `2sqrt(3) - 1` is an irrational number.

Prove that `2 - 3sqrt(5)` is an irrational number.

Prove that `sqrt(5) + sqrt(3)` is irrational.

Given that `sqrt(2)` is irrational, prove that `(5 + 3sqrt(2))` is an irrational number.

Prove that `(2 + sqrt(3))/5` is an irrational number, given that `sqrt(3)` is an irrational number.

Prove that `2 + 5sqrt(3)` is an irrational number, given that `sqrt(3)` is an irrational number.

Prove that `sqrt(2) + sqrt(3)` is irrational.

Prove that `(2 + sqrt(3))` is an irrational number, given that `sqrt(3)` is an irrational number.

Prove that `(5 - 2sqrt(3))` is an irrational number. It is given that `sqrt(3)` is an irrational number.

Prove that `(2 - sqrt(3))/5` is an irrational number, given that `sqrt(3)` is an irrational number.

Prove that `(sqrt(2) + sqrt(3))^2` is an irrational number, given that `sqrt(6)` is an irrational number.

Prove that `(5 sqrt(3) + 2/3)` is an irrational number given that `sqrt(3)` is an irrational number.

Prove that `(4sqrt(2) + 5/3)` is an irrational number given that `sqrt(2)` is an irrational number.

Prove that for any prime positive integer p, `sqrt(p)` is an irrational number.

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 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 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 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 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 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 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.

\[\frac{129}{2^2 \times 5^7}\]

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

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.bar(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....

A rational number in its decimal expansion is 327.7081. What can you say about the prime factors of q, when this number is expressed in the form `p/q`? Given reasons.

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

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

Write whether \[\frac{2\sqrt{45} + 3\sqrt{20}}{2\sqrt{5}}\] on simplification gives a rational or an irrational number.

Find after how many places of decimal the decimal form of the number `27/(2^3. 5^4. 3^2)` will terminate.

Express 429 as the product of its prime factors.

Two positive integers a and b can be written as a = x³y² and b = xy³, x, y are prime numbers. Find LCM (a, b). 

If HCF (336, 54) = 6, find LCM (336, 54).

Show that the number 5 × 11 × 17 + 3 × 11 is a composite number.

If HCF (306, 657) = 9, then LCM (306, 657) ______.

The sum of the exponents of prime factors in the prime factorisation of 250 is ______.

If the prime factorisation of a natural number n is 2⁴ × 3⁴ × 5³ × 7, then the number of consecutive zeros in n, is ______.

If 2³ × 3^a × b × 7 is the prime factorisation of 2520, then 5a + 2b = ______.

If 2520 = 2^a × 3^b × 5^c × 7^d, then a + b – 2c – 3d = ______.

If a and b are two positive co-prime integers such that a = 12b, then HCF (a, 12) = ______.

If two positive integers m and n are expressible in the form m = a²b³ and n = a³b², where a, b are prime numbers, then HCF (m, n) = ______ and LCM (a, b) = ______.

If the HCF and LCM of two positive integers a and b are x and y respectively, then `(x^2y^2)/(a^2b^2)` = ______.

The ratio between the HCF and LCM of 5, 15 and 20 is ______.

The HCF and LCM of two numbers are 33 and 264 respectively. When the first number is completely divided by 2 the quotient is 33. The other number is ______.

Two numbers are in the ratio 21: 17. If their HCF is 5, the numbers are ______ and ______.

Given that LCM (91, 26) = 182, then HCF (91, 26) = ______.

Two numbers are in the ratio 3 : 4 and their LCM is 120. The sum of the numbers is ______.

The LCM of 2x, 5x and 7x and 7x is ______, where x is a positive integer.

A positive integer m when divided by 11 gives remainder 6. If 4m + 5 is divided by 11, the remainder is ______.

The LCM of the smallest prime number and the smallest odd composite number is ______.

The LCM of the smallest prime number and the smallest composite number is ______.

6ⁿ can not end with digit 0 for ______ value of n.

If n is a natural number, then the number of consecutive zeros in 7ⁿ, is ______.

If the least prime factors of two positive integers a and b are 5 and 13 respectively, then the least prime factor of a + b, is ______.

The decimal expansion of `17/8` will terminate after ______ places of decimal.

One of the three consecutive positive integers is always divisible by ______.

6ⁿ cannot end with digit 5 for ______ value of n.

The values of the remainder r, when a positive integer 'a' is divided by 3, are ______.