1. Real Numbers

1.1 Introduction

LEARNING OBJECTIVES.pdf Download

Types of Real Numbers

Prime numbers, Composite numbers and Co-prime numbers

PRIME, COMPOSITE AND COPRIME NUMBERS .pdf Download

Euclid's division algorithm - Explained

1.2 The Fundamental Theorem of Arithmetic

LIST OF PRIME NUMBERS BELOW 200

LIST OF PRIME NUMBERS BELOW 200.pdf Download

DIVISIBILITY RULES

DIVISIBILITY RULES.pdf Download

DIVISIBILITY RULES FOR THE PRIME NUMBERS 13, 17 19 AND 23

DIVISIBILITY RULES FOR THE PRIME NUMBERS 13, 17 19 AND 23.pdf Download

Example 1:

Consider the number 4ⁿ, where n is a natural number. check whether there is any value of n for which 4ⁿ ends with the digit zero.

Example 2, 3, 4.

Find the LCM and HCF of 6 and 20 by the prime factorisation method.
Find the HCF of 96 and 404 by the prime factorisation method. Hence, find their LCM.
Find the HCF and LCM of 6, 72 and 120, using the prime factorisation method.

EXERCISE 1.1

1. Express each number as a product of it's prime factors:

(i) 140    (ii) 156    (iii) 3825  (iv) 5005   (v) 7429

2. Find the LCM and HCF of the following pairs of integers and verify that LCM x HCF = product of the two numbers.

(i) 26 and 91      (ii) 510 and 92    (iii) 336 and 54

3. Find the LCM and HCF of the following integers by applying the prime factorisation method.

(i) 12, 15 and 21    (ii) 17, 23 and 29    (iii) 8, 9 and 25

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

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

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

7. 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 starts 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?

Practice Exercises 1 ─ 5

Can two numbers have 15 as their HCF and 175 as their LCM? Give reasons.
Explain whether 3×12×101 + 4 is a prime number or composite number.
Three numbers are in the ratio 2:5:7 their LCM is 490. find the square root of the largest number.
A warehouse has 1764 kgs of rice and 2352 kgs of sugar. The manager wants to pack them into equal weight packs without any leftover. What is the greatest weight that each pack can hold?
The ratio between LCM and HCF of 5, 15, 20 is …?

Practice Exercises 6 ─ 10

The HCF of 2472, 1284 and a third number N is 12. If their LCM is 2³ × 3² × 5 × 103 × 107, then the number N is …?

Find the LCM of smallest prime and smallest odd composite natural number.

If p and q are two co-prime numbers, then find the HCF and LCM of p and q.

LCM of two numbers is 10 times their HCF. The sum of HCF and LCM is 495. If one number is 90, then find the other number.

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

Practice Exercises 11 ─ 14

Find the largest number which divides 615 and 963 leaving a remainder 6 in each case.
Find the LCM of 2.5, 0.5 and 0.175.
Find the least number which when divided by 16 leaves a remainder 6, when divided by 19 leaves a remainder 9 and when divided by 21 leaves a remainder 11.
If n is a natural number, then 2(5n + 6n) always ends with ….

1.3 Revisiting Irrational Numbers

OPERATIONS BETWEEN RATIONALS AND IRRATIONALS.pdf Download

SQUARE ROOT OF PRIME AND COMPOSITE NUMBERS.pdf Download

Theorem 1.2:

Let p be a prime number. If p divides a², then p divides a, where a is a positive integer.

THEOREM 1.2.pdf Download

Theorem 1.3:

Prove that √2 is irrational

Example 5:

Prove that √3 is irrational.

Example 6:

Show that 5-√3 is irrational.

Example 7:

Show that 3√2 is irrational.

EXERCISE 1.2

1. Prove that √5 is irrational.

2. Prove that 3 + 2√5 is irrational.

3. Prove that the following are irrational.

(i) 1/√2    (ii) 7√5    (iii) 6 + √2

Practice Exercises 15 ─ 19

Check whether 15n can end with digit 0 for any n.
Show that 7n cannot end with digit zero for any natural number n.
Show that any positive odd integer is of the form of 6q+1, or 6q+3 or 6q+5, where q is some integer.
Show that any positive odd integer is of the form of 8q+1, or 8q+3 or 8q+5 or 8q + 7, where q is some integer.
Show that any positive even integer is of the form of 4q or 4q + 2 where q is any integer.

1.4 Summary and Chapter Conclusion

CHAPTER SUMMARY - REAL NUMBERS.pdf Download