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```Chapter 1: Real Numbers Exercise – 1.3

Question: 1

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

1. 420

2. 468

3. 945

4. 7325

Solution:

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

1. 420

420 = 2 × 2 × 3 × 5 × 7

2. 468

468 = 2 × 2 × 3 × 3 × 13

3. 945

945 = 3 × 3 × 3 × 5 × 7

4. 7325

7325 = 5 × 5 × 293

Question: 2

Determine the prime factorization of each of the following positive integer:

1. 20570

2. 58500

3. 45470971

Solution:

To Express: Each of the following numbers as a product of their prime factors.

1. 20570

20570 = 2 × 5 × 11 × 11 × 17

2. 58500

58500 = 2 × 2 × 3 × 3 × 5 × 5 × 5 × 13

3. 45470971

45470971 = 7 × 7 × 13 × 13 × 17 × 17 × 19

Question: 3

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

Solution:

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

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

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

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

Question: 4

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

Solution:

To Check: Whether 6n can end with the digit 0 for any natural number n.

We know that 6n = (2 × 3)n

6n = 2n × 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.
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