# NCERT Mathematics Class 10 Exemplar Ch 1 Real Numbers Part 3 (For CBSE, ICSE, IAS, NET, NRA 2022)

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**EXERCISE 1.3**

1. Show that the square of any positive integer is either of the form or for some integer q.

Answer: The square of any positive integer is either of the form or for some integer q.

2. Show that cube of any positive integer is of the form , or , for some integer m.

Answer: The cube of any positive integer is of the form , or for some integer m.

3. Show that the square of any positive integer cannot be of the form or for any integer q.

Answer: The square of any positive integer cannot be of the form or for any integer q.

4. Show that the square of any positive integer cannot be of the form or for any integer m.

Answer: The square of any positive integer cannot be of the form or for any integer m.

5. Show that the square of any odd integer is of the form , for some integer q.

Answer: For some integer m, the square of any odd integer is of the form .

6. If n is an odd integer, then show that is divisible by 8.

Answer: is divisible by 8.

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

Answer: is even for every positive integer m but not divisible by 4.

8. Use Euclid՚s division algorithm to find the HCF of .

Answer: 63

9. Using Euclid՚s division algorithm, find the largest number that divides and leaving remainders , respectively.

Answer:

10. Prove that is irrational.

Answer: The right hand side is rational number while is irrational. Since, and 5 are prime numbers. is irrational.

11. Show that cannot end with the digit 0 or 5 for any natural number n.

Answer: There is no value of n e N for which ends with digit zero or five.

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

Answer:

13. Write the denominator of the rational number in the form , where m, n are non-negative integers. Hence, write its decimal expansion, without actual division.

Answer:

14. Prove that is irrational, where are primes.

Answer: Number while is irrational, since p and q are prime numbers.

is irrational.