Real Numbers class 10 is an important concept in Mathematics and with our notes and real numbers class 10 extra questions, you can now master this topic easily. Our comprehensive guide covers everything you need to know about Real Numbers Class 10, including helpful notes and extra questions as well as class 10 real numbers MCQ that will enhance your understanding and ensure your success in CBSE Board Exams. With our expert guidance, you’ll gain the skills and confidence you need to achieve in this important subject.

## Context

## Real Numbers Class 10 Notes

### What are real numbers?

**Real numbers** are a type of number that includes all the numbers on the number line. This includes positive and negative numbers, fractions, decimals, and **irrational** numbers like π and √2.

### What are prime numbers?

A **prime number** is a natural number greater than 1 that has only two positive divisors, i.e, 1 and itself.

For example, 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29 are the first ten prime numbers. The number 1 is not a prime number because it only has one positive divisor, which is 1.

### What are composite numbers?

A **composite number** is a number that has more than two positive divisors. For example, 4, 6, 8, 9, 10, 12, 14, 15, 16, and 18 are the first ten composite numbers. These numbers can be divided evenly by at least one positive integer other than 1 and itself.

### What is HCF?

HCF stands for **Highest Common Factor**, which is also known as **Greatest Common Divisor** (GCD). The HCF of two or more integers is the largest positive integer that divides each of the given integers without leaving a remainder.

For example, let’s find the HCF of 12 and 18. The factors of 12 are 1, 2, 3, 4, 6, and 12. The factors of 18 are 1, 2, 3, 6, 9, and 18. The common factors of 12 and 18 are 1, 2, 3, and 6. Out of these common factors, 6 is the largest, so the HCF of 12 and 18 is 6.

### What is LCM?

LCM stands for **Lowest Common Multiple**, which is the smallest positive integer that is a multiple of two or more given integers. In other words, LCM is the smallest number that is divisible by all of the given integers without leaving a remainder.

For example, let’s find the LCM of 4 and 6. The multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, and so on. The multiples of 6 are 6, 12, 18, 24, 30, 36, and so on. The common multiples of 4 and 6 are 12, 24, 36, and so on. Out of these common multiples, 12 is the smallest, so the LCM of 4 and 6 is 12.

## Real Numbers Class 10 HCF and LCM Questions

1. Express 98 as a product of its primes.

2. Find the HCF of numbers 134791, 6341, and 6339 by Euclid’s division algorithm.

3. Find the HCF of 255 and 867 by Euclid’s division algorithm.

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

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

6. Find the HCF (865, 255) using Euclid’s division lemma.

7. Find the LCM of 96 and 360 by using the fundamental theorem of arithmetic.

8. Find the LCM of numbers whose prime factorization is expressible as 3 × 5^{2} and 3^{2} × 7^{2}.

9. Find HCF and LCM of 13 and 17 by prime factorization method.

10. Find HCF of 135 and 224 by both prime factorization and long division methods.

11. Find HCF of 196 and 8220 by both prime factorization and long division methods.

12. Find HCF of 867 and 255 by both prime factorization and long division methods.

13. Find HCF of 4052 and 12576 by Euclid’s division lemma.

14. Find the LCM and HCF of 6 and 20 by the prime factorization method.

15. Find the HCF of 96 and 404 by the prime Factorization method. Hence find their LCM.

16. Find the HCF and LCM of 6,72, and 120 using the Prime factorization method.

17. Find the largest number that will divide 398, 436, and 542 leaving remainders 7, 11, and 15 respectively.

18. If the HCF of 408 and 1032 is expressible in the form 1032 × 2 + 408 × p, then find the value of p.

19. HCF and LCM of two numbers are 9 and 459 respectively. If one of the numbers is 27, find the other number.

20. Find the largest number which divides 70 and 125 leaving the remainder 5 and 8 respectively.

21. Three bells toll at intervals of 9, 12, and 15 minutes respectively. If they start tolling together, after what time will they next toll together?

22. Two tankers contain 850 liters and 680 liters of petrol. Find the maximum capacity of a container that can measure the petrol of each tanker the exact number of times.

23. Three alarm clocks ring at intervals of 4, 12, and 20 minutes respectively. If they start ringing together, after how much time will they next ring together?

24. In a school, there are two Sections A and B of class X. There are 48 students in Section A and 60 students in Section B. Determine the least number of books required for the library of the school so that the books can be distributed equally among all students of each Section.

25. There are 104 students in class X and 96 students in class IX in a school. In a house examination, the students are to be evenly seated in parallel rows such that no two adjacent rows are of the same class. (2013)

(a) Find the maximum number of parallel rows of each class for the seating arrange¬ment.

(b) Also, find the number of students of class IX and also of class X in a row.

(c) What is the objective of the school administration behind such an arrangement?

26. Dudhnath has two vessels containing 720 ml and 405 ml of milk respectively. Milk from these containers is poured into glasses of equal capacity to their brim. Find the minimum number of glasses that can be filled.

27. Amita, Sneha, and Raghav start preparing cards for all persons of an old age home. In order to complete one card, they take 10, 16, and 20 minutes respectively. If all of them started together, after what time will they start preparing a new card together?

28. If two positive integers x and y are expressible in terms of primes as x = p^{2}q^{3} and y = p^{3}q, what can you say about their LCM and HCF? Is LCM a multiple of HCF?

29. Find the HCF and LCM of 306 and 657 and verify that LCM × HCF = Product of the two numbers.

## Real Numbers Class 10 Prime and Composite Numbers Questions

1. Find the prime factorization of the denominator of the rational number expressed as 6.12¯ in simplest form.

2. Explain why (17 × 5 × 11 × 3 × 2 + 2 × 11) is a composite number.

3. Check whether 4n can end with the digit 0 for any natural number n.

4. Explain why ( 7 × 11× 13+13) is a composite number.

5. Exp[lain why (7 × 6 × 5 × 4 × 3 × 2 × 1 +5) is a composite number.

## Real Numbers Class 10 Proving Irrational Questions

1. Prove that √5 is irrational and hence show that 3 + √5 is also irrational.

2. Show that 3√7 is an irrational number.

3. Prove that 2 + 3√5 is an irrational number.

4. Prove that 3 + 2√3 is an irrational number.

5. Prove that 3 + 2√5 is irrational.

## Real Numbers Class 10 Miscellaneous Questions

1. The decimal expansion of the rational number 43/2^{4}5^{3} will terminate after how many places of decimals?

2. Write the decimal form of 129/2^{7}5^{7}7^{5}

3. Check whether 4^{n} can end with the digit 0 for any natural number n.

4. Show that one and only one out of n, (n + 1) and (n + 2) is divisible by 3, where n is any positive integer.

6. Show that any positive odd integer is of form 41 + 1 or 4q + 3 where q is a positive integer.

## Class 10 Real Numbers MCQ

**1: The product of a rational and irrational number is**

(a) rational

(b) irrational

(c) both of above

(d) none of above

**2: The decimal expansion of 22/7 is**

(a) Terminating

(b) Non-terminating and repeating

(c) Non-terminating and Non-repeating

(d) None of the above

**3: Which of the following is not irrational?**

(a) (2 – √3)2

(b) (√2 + √3)2

(c) (√2 -√3)(√2 + √3)

(d) 2√7/7

4: **For some integer n, the odd integer is represented in the form of:**

(a) n

(b) n + 1

(c) 2n + 1

(d) 2n

**5: The sum of a rational and irrational number is**

(a) rational

(b) irrational

(c) both of above

(d) none of above

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

(a) a < r < b

(b) 0 < r ≤ b

(c) 1 < r < b

(d) 0 ≤ r < b

**7: If LCM (77, 99) = 693, then HCF (77, 99) is**

(a) 11

(b) 7

(c) 9

(d) 22

**8: ‘π’ is a**

(a) natural number

(b) rational number

(c) irrational number

(d) rational or irrational

**9: HCF of 26 and 91 is:**

(a) 15

(b) 13

(c) 19

(d) 11

**10:** **Which of the following is not irrational?**

(a) (3 + √7)

(b) (3 – √7)

(c) (3 + √7) (3 – √7)

(d) 3√7

**11:** **The multiplication of two irrational numbers is:**

(a) irrational number

(b) rational number

(c) Maybe rational or irrational

(d) None

**12:** **The largest number that divides 70 and 125, which leaves the remainders 5 and 8, is:**

(a) 65

(b) 15

(c) 25

(d) 13

**13:** **The least number that is divisible by all the numbers from 1 to 5 is:**

(a) 70

(b) 60

(c) 80

(d) 90

**14: For some integer m, every even integer is of the form**

(a) m

(b) m + 1

(c) 2m

(d) 2m + 1

**15:** **If the HCF of 65 and 117 is expressible in the form 65m – 117, then the value of m is**

(a) 2

(b) 5

(c) 1

(d) 3

**16: If LCM (77, 99) = 693, then HCF (77, 99) is**

(a) 11

(b) 7

(c) 9

(d) 22

**17: If two positive integers A and B can be expressed as A = xy ^{3} and B = x^{4}y^{2}z; x, y being prime numbers then HCF (A, B) is**

(a) x^{4}y^{3}z

(b) x^{4}y²z

(c) x^{4}y^{3}

(d) xy^{2}

**18: The product of two consecutive natural numbers is always**

(a) prime number

(b) even number

(c) odd number

(d) even or odd

**19: If the HCF of 408 and 1032 is expressible in the form 1032 x 2 + 408 × p, then the value of p is**

(a) 5

(b) 4

(c) -5

(d) -4

**20: If HCF (16, y) = 8 and LCM (16, y) = 48, then the value of y is**

(a) 24

(b) 16

(c) 8

(d) 48

We hope this article will help you a lot to understand and master Real Numbers Class 10. Real numbers class 10 extra questions are very important to develop strong concepts moreover Class 10 real Numbers MCQ adds up an extra level of mastery to your concepts from the examination point of view.