NCERT solutions for Mathematics [English] Class 10 chapter 1 - Real Numbers [2018 edition]

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

Using Euclid's division algorithm, find the H.C.F. of (iii) 867 and 255

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

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?

 Use Euclid's Division Algorithm to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m.

Use Euclid's Division Algorithm to show that the cube of any positive integer is either of the 9m, 9m + 1 or 9m + 8 for some integer m

Express the number as a product of its prime factor:

140

Express the number as a product of its prime factor:

156

Express the number as a product of its prime factor:

3825

Express the number as a product of its prime factor:

5005

Express the number as a product of its prime factor:

7429

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

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

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

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

There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction. After how many minutes will they meet again at the starting point?

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

Prove that 3 + 2`sqrt5` is irrational.

Prove that the following is irrational:

`1/sqrt2`

Prove that the following is irrational:

`7sqrt5`

Prove that the following are irrational.

`6+sqrt2`

State whether the following rational number will have a terminating decimal expansion or a non-terminating repeating decimal expansion:

`13/3125`

State whether the following rational number will have a terminating decimal expansion or a non-terminating repeating decimal expansion:

`17/8`

State whether the following rational number will have a terminating decimal expansion or a non-terminating repeating decimal expansion:

`64/455`

State whether the following rational number will have a terminating decimal expansion or a non-terminating repeating decimal expansion:

`15/1600`

State whether the following rational number will have a terminating decimal expansion or a non-terminating repeating decimal expansion:

`29/343`

State whether the following rational number will have a terminating decimal expansion or a non-terminating repeating decimal expansion:

 `23/(2^3xx5^2)`

State whether the following rational number will have a terminating decimal expansion or a non-terminating repeating decimal expansion:

`129/(2^2xx5^7xx7^5)`

State whether the following rational number will have a terminating decimal expansion or a non-terminating repeating decimal expansion:

`6/15`

State whether the following rational number will have a terminating decimal expansion or a non-terminating repeating decimal expansion:

`35/50`

State whether the following rational number will have a terminating decimal expansion or a non-terminating repeating decimal expansion:

 `77/210`

Write down the decimal expansion of the following number  which have terminating decimal expansion.

`13/3125`

Write down the decimal expansion of the following number  which have terminating decimal expansion.

 `17/8`

Write down the decimal expansion of the following number  which have terminating decimal expansion.

`64/455`

Write down the decimal expansion of the following number  which have terminating decimal expansion.

`15/1600`

Write down the decimal expansion of the following number  which have terminating decimal expansion.

`29/343`

Write down the decimal expansion of the following number  which have terminating decimal expansion.

`23/(2^3xx5^2)`

Write down the decimal expansion of the following number  which have terminating decimal expansion.

`129/(2^2xx5^7xx7^5)`

Write down the decimal expansion of the following number  which have terminating decimal expansion.

`6/15`

Write down the decimal expansion of the following number  which have terminating decimal expansion.

`35/50`

Write down the decimal expansion of the following number which has a terminating decimal expansion.

`77/210`

The following real number have decimal expansions as given below. In the following case, decide whether it is  rational or not. If it is rational, and of the form p/q what can you say about the prime factors of q?

43.123456789