NCERT Class X Chapter 1: Real Numbers Exercise 1.1 Question 6

Question:

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.

Given:

The two expressions:

• \(7 \times 11 \times 13 + 13\)
• \(7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5\)

To Find:

Show, with step-by-step reasoning, that both expressions are composite numbers.

Formula:

A composite number is a whole number greater than 1 that has more than two distinct positive divisors (i.e., it is not prime). To show a number is composite, we factorize it or show it has divisors other than 1 and itself.

Solution:

Step 1: Consider the first expression: \(7 \times 11 \times 13 + 13\).

$$ 7 \times 11 \times 13 + 13 $$

Step 2: Factor out 13 from both terms.

$$ = (7 \times 11 \times 13) + 13 \\ = 13 \times (7 \times 11) + 13 \times 1 \\ = 13 \times (7 \times 11 + 1) $$

Step 3: Simplify inside the bracket.

$$ 7 \times 11 = 77 \\ 77 + 1 = 78 \\ \Rightarrow 13 \times 78 $$

Step 4: Calculate the value.

$$ 13 \times 78 = 1014 $$

Step 5: Check if 1014 is composite.

1014 has divisors 13 and 78, which are both greater than 1 and less than 1014. So, 1014 is a composite number.

Step 6: Consider the second expression: \(7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5\).

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

Step 7: Recognize and calculate the factorial.

$$ 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 7! = 5040 $$

Step 8: Add 5 to the factorial value.

$$ 5040 + 5 = 5045 $$

Step 9: Check divisibility by 5 (since the last digit is 5).

$$ 5045 \div 5 = 1009 \\ \Rightarrow 5045 = 5 \times 1009 $$

Step 10: Check if 5045 is composite.

5045 has divisors 5 and 1009, both greater than 1 and less than 5045. Therefore, 5045 is a composite number.

Result:

Both \(7 \times 11 \times 13 + 13\) and \(7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5\) are composite numbers because each can be factored into the product of numbers other than 1 and itself.

Next question solution:

NCERT Class X Chapter 1: Real Numbers Exercise 1.1 7

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