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

There are two types of numbers, namely, prime and composite. Prime numbers have only two factors namely 1 and the number itself whereas composite numbers have factors other than 1 and itself. It can be observed that, 7 × 11 × 13 + 13 = 13 × ( 7 × 11 + 1 ) = 13 × ( 77 + 1 ) = 13 × 78 = 13 × 13 × 6 The given expression has 6 and 13 as its factors. Therefore, it is a composite number. 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 = 5 × ( 7 × 6 × 4 × 3 × 2 × 1 + 1 ) = 5 × ( 1008 + 1 ) = 5 × 1009 . 1009 cannot be factorized further. Therefore, the given expression has 5 and 1009 as its factors. Hence, it is a composite number.

Prove that 5 is irrational.

Let us assume, on the contrary, that 5 is a rational number. Therefore, we can find two co-prime integers a , b ( b ≠ 0 ) such that 5 = a b . 5 = a b ⇒ a = 5 b ⇒ a 2 = 5 b 2 Therefore, a 2 is divisible by 5 , then a is also divisible by 5 . So, a 2 = 5 k 2 = 5 5 k 2 = 5 b 2 ⇒ b 2 = 5 k 2 This means that b 2 is divisible by 5 and hence, b is divisible by 5 . This implies that a and b have 5 as a common factor. And this is a contradiction to the fact that a and b are co-prime. So, our assumption that 5 is rational is wrong. Hence, 5 cannot be a rational number. Therefore, 5 is irrational.

Prove that 3 + 2 5 is irrational.

Let us assume, on the contrary, that 3 + 2 5 is rational. Therefore, we can find two integers a , b ( b ≠ 0 ) such that 3 + 2 5 = a b 2 5 = a b - 3 5 = 1 2 a b - 3 Since a and b are integers, 1 2 a b - 3 will also be rational and therefore, 5 is rational. This contradicts the fact that 5 is irrational. Hence, our assumption that 3 + 2 5 is rational is false. Therefore, 3 + 2 5 is irrational.

Prove that the following is irrational. 1 2

Given : 1 2 Let us assume that 1 2 is rational. Therefore, we can consider co-primes as a , b ( b ≠ 0 ) such that 1 2 = a b 2 = b a b a is rational as a and b are integers. Therefore, 2 is rational. This contradicts the fact that 2 is irrational. Hence, our assumption is false and 1 2 is irrational.

Prove that the following are irrational. 7 5

Given, 7 5 Let us assume that 7 5 is rational. Therefore, we can find two integers a , b ( b ≠ 0 ) such that 7 5 = a b ∴ 5 = a 7 b a 7 b is rational as a and b are integers. Therefore, 5 should be rational. This contradicts the fact that 5 is irrational. Therefore, our assumption that 7 5 is rational is false. Hence, 7 5 is irrational.

Prove that the following is irrational. 6 + 2

Let 6 + 2 be rational. Therefore, we can find two integers a , b b ≠ 0 such that, 6 + 2 = a b 2 = a b - 6 Since a and b are integers, a b - 6 is a rational number and hence, 2 should be rational. This contradicts the fact that 2 is irrational. Therefore, our assumption is false and hence, 6 + 2 is irrational.

E M B I B E

MATHEMATICS Textbook for Class X>Chapter 1 - Real Numbers>Exercise 1.2>Q 5

EASY

10th Assam Board

IMPORTANT

Earn 100

Check whether $6^{n}$ can end with the digit $0$ for any natural number $n$ .

Important Questions on Real Numbers

MEDIUM

10th Assam Board

IMPORTANT

MATHEMATICS Textbook for Class X>Chapter 1 - Real Numbers>Exercise 1.2>Q 6

Explain why

7 \times 11 \times 13 + 13

and

7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5

are composite numbers.

MEDIUM

10th Assam Board

IMPORTANT

MATHEMATICS Textbook for Class X>Chapter 1 - Real Numbers>Exercise 1.2>Q 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 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?

HARD

10th Assam Board

IMPORTANT

MATHEMATICS Textbook for Class X>Chapter 1 - Real Numbers>Exercise 1.3>Q 1

Prove that

\sqrt{5}

is irrational.

MEDIUM

10th Assam Board

IMPORTANT

MATHEMATICS Textbook for Class X>Chapter 1 - Real Numbers>Exercise 1.3>Q 2

Prove that

3 + 2 \sqrt{5}

is irrational.

MEDIUM

10th Assam Board

IMPORTANT

MATHEMATICS Textbook for Class X>Chapter 1 - Real Numbers>Exercise 1.3>Q 3

Prove that the following is irrational.

\frac{1}{\sqrt{2}}

MEDIUM

10th Assam Board

IMPORTANT

MATHEMATICS Textbook for Class X>Chapter 1 - Real Numbers>Exercise 1.3>Q 3

Prove that the following are irrational.

7 \sqrt{5}

MEDIUM

10th Assam Board

IMPORTANT

MATHEMATICS Textbook for Class X>Chapter 1 - Real Numbers>Exercise 1.3>Q 3

Prove that the following is irrational.

6 + \sqrt{2}

EASY

10th Assam Board

IMPORTANT

MATHEMATICS Textbook for Class X>Chapter 1 - Real Numbers>Exercise 1.4>Q 1

Without actually performing the long division state whether the following rational number will have a terminating decimal expansion or a non-terminating repeating decimal expansion.