RD Sharma Solutions Class 10 Real Numbers Exercise 1.3

Exercise 1.3

Q.1: Express each of the following integers as a product of its prime.

1. 420

2. 468

3. 945

4. 7325

Sol:

To express: each of the following numbers as a product of their prime factors

1. 420

420 = 2 x 2 x 3 x 5 x 7

2. 468

468 = 2 x 2 x 3 x 3 x 13

3. 945

945 = 3 x 3 x 3 x 5 x 7

4. 7325

7325 = 5 x 5 x 293

Q.2: Determine the prime factorization of each of the following positive integer :

1. 20570

2. 58500

3. 45470971

Sol:

TO EXPRESS: each of the following numbers as a product of their prime factors.

1. 20570

20570 = 2 x 5 x 11 x 11 x 17

2. 58500

58500 = 2 x 2 x 3 x 3 x 5 x 5 x 5 x 13

3. 45470971

45470971 = 7 x 7 x 13 x 13 x 17 x 17 x 19

Q.3: 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.

Sol:

Explanation:

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.

We can see that both the numbers have common factor 7 and 1.

7 x 11 x 13 + 13 = (77 + 1) x 13 = 78 x 13

7 x 6 x 5 x 4 x 3 x 2 x 1 + 5 = (7 x 6 x 4 x 3 x 2 + 1) x 5 = 1008 x 5

And we know that composite numbers are those numbers which have at least one more factor other than 1.

Hence after simplification we see that both numbers are even and therefore the given two numbers are composite numbers

Q.4: Check whether 6ⁿ can end with the digit 0 for any natural number n.

Sol:

TO CHECK: Whether 6ⁿ can end with the digit 0 for any natural number n.

We know that 6ⁿ = (2 x 3)ⁿ

6ⁿ = 2ⁿ x 3ⁿ

Therefore, prime factorization of 6ⁿ does not contain 5 and 2 as a factor together. Hence 6ⁿ can never end with the digit 0 for any natural number n.