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

What do you mean by Euclid’s division lemma?

A number when divided by 61 gives 27 as quotient and 32 as remainder. Find the number.

By what number should 1365 be divided to get 31 as quotient and 32 as remainder?

Using Euclid’s division algorithm, find the HCF of 612 and 1314.

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

Using Euclid’s division algorithm, find the HCF of 4052 and 12576.

By using Euclid’s algorithm, find the largest number which divides 650 and 1170.

Find the HCF of the smallest prime number and the smallest composite number.

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

Prove that if x and y are both odd positive integers, then x² + y² is even but not divisible by 4.

Use Euclid’s algorithm to find the HCF of 1190 and 1445. Express the HCF in the form 1190m + 1445n.

Use Euclid’s division algorithm to find the HCF of 441, 567, 693.

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

Express 429 as a product of its prime factors.

Express 5005 as a product of its prime factors.

Express 2431 as a product of its prime factors.

Using prime factorisation, find the HCF and LCM of 36, 84. In case verify that HCF × LCM = product of given numbers.

Using prime factorisation, find the HCF and LCM of 23, 31. In case verify that HCF × LCM = product of given numbers.

Using prime factorisation, find the HCF and LCM of 96, 404. In case verify that HCF × LCM = product of given numbers.

Using prime factorisation, find the HCF and LCM of 144, 198. In case verify that HCF × LCM = product of given numbers.

Using prime factorisation, find the HCF and LCM of 396, 1080. In case verify that HCF × LCM = product of given numbers.

Using prime factorisation, find the HCF and LCM of 1152, 1664. In case verify that HCF × LCM = product of given numbers.

Using prime factorisation, find the HCF and LCM of 8, 9, 25.

Using prime factorisation, find the HCF and LCM of 12, 15, 21.

Using prime factorisation, find the HCF and LCM of 17, 23, 29.

Using prime factorisation, find the HCF and LCM of 24, 36, 40.

 Using prime factorisation, find the HCF and LCM of 30, 72, 432.

Using prime factorisation, find the HCF and LCM of 21, 28, 36, 45.

Find HCF and LCM of 404 and 96 and verify that HCF × LCM = Product of the two given numbers.

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

The HCF of two numbers is 23 and their LCM is 1449. If one of the numbers is 161, find the other.

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

The HCF of two numbers a and b is 5 and their LCM is 200. Find the product ab.

The LCM of two numbers is 9 times their HCF. The sum of LCM and HCF is 500. Find their HCF.

The HCF of two numbers is 18 and their product is 12960. Find their LCM.

Can two numbers have 15 as their HCF and 175 as their LCM? Give reason.

Find the simplest form of `69/92`.

Find the simplest form of `368 /496`.

Find the simplest form of `473/645`.

Find the simplest form of `1095/1168`. 

Find the largest number which divides 438 and 606, leaving remainder 6 in each case.

Find the largest number which divides 320 and 457, leaving remainders 5 and 7 respectively.

Find the least number which when divides 35, 56 and 91 leaves the same remainder 7 in each case.

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

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

Find the greatest number of four digits which is exactly divisible by 15, 24 and 36.

Find the largest four-digit number which when divided by 4, 7 and 13 leaves a remainder of 3 in each case.

Find the least number which should be added to 2497 so that the sum is exactly divisible by 5, 6, 4 and 3.

Find the greatest number that will divide 43, 91 and 183 so as to leave the same remainder in each case.

Find the least number which when divided by 20, 25, 35 and 40 leaves remainders 14, 19, 29 and 34 respectively.

In a seminar, the number of participants in Hindi, English and mathematics are 60, 84 and 108 respectively. Find the minimum number of rooms required, if in each room, the same number of participants are to be seated and all of them being in the same subject . 

Three sets of English, Mathematics and Science books containing 336, 240 and 96 books respectively have to be stacked in such a way that all the books are stored subject-wise and the height of each stack is the same. How many stacks will be there?

Three pieces of timber 42 m, 49 m and 63 m long have to be divided into planks of the same length. What is the greatest possible length of each plank? How many planks are formed?

Find the greatest possible length which can be used to measure exactly the lengths 7 m, 3 m 85 cm and 12 m 95 cm.

Find the maximum number of students among whom 1001 pens and 910 pencils can be distributed in such a way that each student gets the same number of pens and the same number of pencils.

Find the least number of square tiles required to pave the ceiling of a room 15 m 17 cm long and 9 m 2 cm broad.

Three measuring rods are 64 cm, 80 cm and 96 cm in length. Find the least length of cloth that can be measured an exact number of times, using any of the rods.

An electronic device makes a beep after every 60 seconds. Another device makes a beep after every 62 seconds. They beeped together at 10 a.m. At what time will they beep together at the earliest?

The traffic lights at three different road crossings change after every 48 seconds, 72 seconds and 108 seconds respectively. If they all change simultaneously at 8 a.m. then at what time will they again change simultaneously?

Six bells commence tolling together and toll at intervals of 2, 4, 6, 8, 10, 12 minutes respectively. In 30 hours, how many times do they toll together?

Show that 6ⁿ can never end with 0 for any natural number n.

Without actual division, show that the following rational numbers is a terminating decimal. Express in decimal form.

`23/((2^3 xx 5^2))`

Without actual division, show that the following rational numbers is a terminating decimal. Express in decimal form.

`24/125`

Without actual division, show that the following rational numbers is a terminating decimal. Express in decimal form. 

`171/800`

Without actual division, show that the following rational numbers is a terminating decimal. Express in decimal form. 

`15/1600`

Without actual division, show that the following rational numbers is a terminating decimal. Express in decimal form.

`17 /320`

Without actual division, show that the following rational numbers is a terminating decimal. Express in decimal form.

`19/3125`

Without actual division, show that the following rational numbers is a non-terminating repeating decimal. 

`11/((2^3 xx 3))`

Without actual division, show that the following rational numbers is a non-terminating repeating decimal.

`73/((2^3 xx 3^3 xx 5))`

Without actual division, show that the following rational numbers is a non-terminating repeating decimal. 

`129/((2^2 xx 5^7 xx 7^5))`

Without actual division, show that the following rational numbers is a non-terminating repeating decimal.

`9/35`

Without actual division, show that the following rational numbers is a non-terminating repeating decimal. 

`77/210`

Without actual division, show that the following rational numbers is a non-terminating repeating decimal. 

`32/147`

Without actual division, show that the following rational numbers is a non-terminating repeating decimal. 

`29/343`

Without actual division, show that the following rational numbers is a non-terminating repeating decimal. 

`64/455`

Express the following as a rational number in its simplest form: 

`0.bar(8)`

Express the following as a rational number in its simplest form:

`2.bar(4)`

Express the following as a rational number in its simplest form:

`0.bar(24)`

Express the following as a rational number in its simplest form:

`0.bar(12)`

Express the following as a rational number in its simplest form: 

`2.bar(24)`

Express the following as a rational number in its simplest form: 

`0.bar(365)`

Define rational numbers. 

Define irrational numbers.

Define real numbers.

Classify the following numbers as rational or irrational: 

`22/7`

Classify the following numbers as rational or irrational:

3.1416

Classify the following numbers as rational or irrational:

π

Classify the following numbers as rational or irrational:

`3.bar(142857)`

Classify the following numbers as rational or irrational:

5.636363...

Classify the following numbers as rational or irrational:

2.040040004...

Classify the following numbers as rational or irrational:

1.535335333...

Classify the following numbers as rational or irrational:

3.121221222...

Classify the following numbers as rational or irrational:

`sqrt(21)`

Classify the following numbers as rational or irrational:

`root(3)(3)`

Find a rational number between `sqrt(2)` and `sqrt(3)`.

Prove that `sqrt(6)` is an irrational number.

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

Prove that `(4 - sqrt(3))` is an irrational number, given that `sqrt(3)` is an irrational number.

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

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

Prove that `((3 - 4sqrt(2)))/7` is an irrational number, given that `sqrt(2)` 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 `5sqrt(2)` is an irrational number, given that `sqrt(2)` is an irrational number.

Prove that `1/sqrt(3)` is irrational, given that `sqrt(3)` is irrational.

Prove that `2/sqrt(7)` is irrational, given that `sqrt(7)` is irrational.

Prove that `3/sqrt(5)` is irrational, given that `sqrt(5)` is irrational.

Prove that `(sqrt(2) + sqrt(5))` is irrational.

Give an example of two irrationals whose sum is rational.

Give an example of two irrationals whose product is rational.

State whether the given statement is true or false: 

The sum of two rationals is always rational

State whether the given statement is true or false:

The product of two rationals is always rational

State whether the given statement is true or false:

The sum of two irrationals is always an irrational

State whether the given statement is true or false:

The product of two irrationals is always an irrational

State whether the given statement is true or false:

The sum of a rational and an irrational is irrational

State whether the given statement is true or false:

The product of a rational and an irrational is irrational

State fundamental theorem of arithmetic.

Express 360 as product of its prime factors.

If a and b are two prime numbers then find HCF(a, b).

If a and b are two prime numbers then find LCM(a, b).

If the product of two numbers is 1050 and their HCF is 25, find their LCM.

What is a composite number?

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

If the rational number `a/b` has a terminating decimal expansion, what is the condition to be satisfied by b?

Simplify: `(2sqrt(27) + 3sqrt(12))/(4sqrt(3))`.

Write the decimal expansion of `73/((2^4 xx 5^3))`.

Show that there is no value of n for which (2ⁿ × 5ⁿ) ends in 5.

Is it possible to have two numbers whose HCF is 25 and LCM is 520?

Give an example of two irrationals whose sum is rational.

Give an example of two irrationals whose product is rational.

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

The LCM of two numbers is 1200. Show that the HCF of these numbers cannot be 500. Why ?

Express `0.bar(4)` as a rational number in simplest form.

Express `0.bar(23)` as a rational number in simplest form.

Explain why 0.15015001500015 ... is an irrational number.

Show that `sqrt(2)/3` is irrational.

Write a rational number between `sqrt(3)` and 2.

Explain why `3.bar(1416)` is a rational number.

Which of the following is a pair of co-primes?

If a = (2² × 3³ × 5⁴) and b = (2³ × 3² × 5) then HCF(a, b) = ?

HCF of (2³ × 3² × 5), (2² × 3³ × 5²) and (2⁴ × 3 × 5³ × 7) is ______.

LCM of (2³ × 3 × 5) and (2⁴ × 5 × 7) is ______.

The HCF of two numbers is 27 and their LCM is 162. If one of the numbers is 54, what is the other number?

The product of two numbers is 1600 and their HCF is 5. The LCM of the numbers is ______.

What is the largest number that divides each one of 1152 and 1664 exactly?

What is the largest number that divides 70 and 125, leaving remainders 5 and 8 respectively?

What is the largest number that divides 245 and 1029, leaving remainder 5 in each case?

The simplest form of `1095/1168` is ______.

Euclid’s division lemma states that for any positive integers a and b, there exist unique integers q and r such that a = bq + r, where r must satisfy

A number when divided by 143 leaves 31 as remainder. What will be the remainder when the same number is divided by 13?

Which of the following is an irrational number?

π is ______.

`2.bar(35)` is ______.

2.13113111311113 ... is ______.

The number 3.24636363... is ______.

Which of the following rational numbers is expressible as a terminating decimal?

The decimal expansion of the rational number will terminate `37/(2^2 xx 5)` will terminate after

The decimal expansion of the number `14753/1250` will terminate after

The number 1.732 is ______.

a and b are two positive integers such that the least prime factor of a is 3 and the least prime factor of b is 5. Then, the least prime factor of (a + b) is ______.

`sqrt(2)` is ______.

`1/sqrt(2)` is ______.

`(2 + sqrt(2))` is ______.

What is the least number that is divisible by all the natural numbers from 1 to 10 (both inclusive)?

The decimal representation of `71/150` is ______.

Which of the following has a terminating decimal expansion?

On dividing a positive integer n by 9, we get 7 as remainder. What will be the remainder if (3n – 1) is divided by 9?

`0.bar(68) + 0.bar(73)` = ?

Show that any number of the form 4ⁿ, n ∈ N can never end with the digit 0.

The HCF of two numbers is 27 and their LCM is 162. If one of the number is 81, find the other.

Examine whether `17/30` is a terminating decimal.

Find the simplest form of `148/185`.

Which of the following numbers are irrational?

Prove that `(4 + 3sqrt(5))` is irrational.

Find the HCF and LCM of 12, 15, 18, 27.

Give an example of two irrationals whose sum is rational.

Give prime factorisation of 4620.

Find the HCF of 1008 and 1080 by prime factorisation method.

Find the HCF and LCM of `8/9, 10/27` and `16/81`.

Find the largest number which divides 546 and 764, leaving remainders 6 and 8 respectively.

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

Show that every positive odd integer is of the form (4q + 1) or (4q + 3) for some integer q.

Show that one and only one out of n; n + 2 or n + 4 is divisible by 3, where n is any positive integer.

Show that `(4 + 3sqrt(2))` is irrational.