Question

explain why 7*11*13+13 and 7*6*5*4*3*2*1+5 are composite numbers

Answer 1

By definition, 

A composite number is a positive integer that has a factor other than 1 and itself. Now considering your numbers,

7×11×13+13 may be written as, $13 * [(7 * 11) + 1]$ i.e. 13 * (78). So other than 1 and the number itself, 13 and 78 are also the factors of the number. Further, 78= 39 x 2. So, 39 and 2 are also it's factors. So this number is definitely not prime. Hence its composite number.

Similarly, 7×6×5×4×3×2×1+5 can be written as $5*[(7*6*4*3*2*1) + 1]$ , i.e. 5 * (1009). So, other than the number and 1, it have 5 and 1009 as it's factors too. So it is also a composite number.

Answer 2

1st case
7*11*13+13 = 13(7*11+1) = 13*78 = 13*2*3*13
Since it has more than two prime factors, it is a composite number.

Similarly,

2nd case
7*6*5*4*3*2*1+5 = 5(7*6*4*3*2*1+1) = 5*1009
Since it has more than two prime factors, it is a composite number.