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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.

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So, basically there are two types of numbers i.e., prime numbers and composite numbers. 

Understanding that, 

Prime numbers are those numbers having 1 and the number itself as factors. And, 

Composite numbers are those numbers having factors other than 1 and itself. 

It’s seen that, 

7 x 11 x 13 + 13 = 13 x (7 x 11 + 1) [taking 13 out as common] 

= 13 x (77 + 1) 

= 13 x 78 

= 13 x 13 x 6 

So, the given expression has 6 and 13 as its factors. Therefore, we can conclude that it is a composite number. 

Similarly, 

7 x 6 x 5 x 4 x 3 x 2 x 1 + 5 

= 5 x (7 x 6 x 4 x 3 x 2 x 1 + 1) [taking 5 out- common] 

= 5 x (1008 + 1) 

= 5 x 1009 

Since, 1009 is a prime number the given expression has 5 and 1009 as its factors other than 1 and the number itself. 

Hence, it is also a composite number.

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