Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers
Solution:
Expression 1: 7 × 11 × 13 + 137
Factor out the common term
7 × 11 × 13 + 13 = 13 × (7 × 11 + 1)
Simplify the expression inside parentheses
7 × 11 + 1 = 77 + 1 = 78
⇒ 13 × 78
Since 13 and 78 are both greater than 1, the number is composite.
Expression 2: 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5
Recognize the factorial term
7! + 5
where 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 504.
Factor out the common term
7! + 5 = 5040 + 5 = 504
Check divisibility
Since 5045 can be factored as:
5045 = 5 × 1009
where 5 and 1009 are greater than 1, this means 5045 is composite.