A composite number is a positive integer that has at least one positive divisor other than one or itself. In simpler terms, it can be divided evenly by numbers other than one and itself.
For the first expression, 7 x 11 x 13, we can see that:
- The numbers 7, 11, and 13 are all prime numbers, meaning they can only be divided evenly by 1 and themselves.
- However, when we multiply these prime numbers together, we create a new number:
- 7 x 11 = 77
- 77 x 13 = 1001
The final product, 1001, is a composite number because it can be factored into the prime numbers 7, 11, and 13, confirming it has divisors other than 1 and itself.
Now, looking at the second expression, 7 x 6 x 5 x 4 x 3 x 2 x 15:
- First, let’s break it down:
- 7 is prime.
- 6 can be factored into 2 x 3.
- 5 is prime.
- 4 can be factored into 2 x 2.
- 3 is prime.
- 2 is prime.
- 15 can be factored into 3 x 5.
If we consider the whole product:
7 x 6 x 5 x 4 x 3 x 2 x 15 = 7 x (2 x 3) x 5 x (2 x 2) x 3 x 2 x (3 x 5).
This expands to show that we are essentially multiplying several numbers, some of which are repeated. The final product combines multiple factors, making it evident that this number has divisors other than one and itself. Therefore, it is also a composite number.
In summary, both numbers are composite because they each can be divided into smaller factors, confirming they have multiple divisors.