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

Solution:

To solve this question, recall that:

• Prime numbers are whole numbers whose only factors are 1 and the number itself.
• Composite numbers are positive integers that have factors other than 1 and themselves.

Now, simplify 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5.

On simplifying them, we find that both the numbers have more than two factors. So, if the number has more than two factors, it will be composite.

It can be observed that,

7 × 11 × 13 + 13 = 13 (7 × 11 + 1)

= 13(77 + 1)

= 13 × 78

= 13 × 13 × 6 × 1

= 13 × 13 × 2 × 3 × 1

The given number has 2, 3, 13, and 1 as its factors. Therefore, it is a composite number.

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

= 5 × (1008 + 1)

= 5 × 1009 × 1

1009 cannot be factorized further. Therefore, the given expression has 5,1009 and 1 as its factors. Hence, it is a composite number.

☛ Check: NCERT Solutions Class 10 Maths Chapter 1

Video Solution:

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

NCERT Solutions Class 10 Maths Chapter 1 Exercise 1.2 Question 6

Summary:

The numbers 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers since on simplification the numbers have (2,3,13 and 1) and (5,1009 and 1) as their factors respectively.

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