Explain why $7 \times 11 \times 13 + 13$ and $7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5$ are composite numbers

Explain why $7 \times 11 \times 13 + 13$ and $7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5$ are composite numbers

To prove that the given expressions are composite numbers, we show that they have factors other than 1 and themselves.

### First Expression: $7 \times 11 \times 13 + 13$

Factor out the common term $13$:

$ 7 \times 11 \times 13 + 13 = 13 \times (7 \times 11 + 1) $

Simplify inside the parentheses:

$ = 13 \times (77 + 1) $

$ = 13 \times 78 $

Since $13$ and $78$ are both greater than 1, the expression is composite.

### Second Expression: $7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5$

Rewrite as:

$ 7! + 5 $

Factor out $5$:

$ 7! + 5 = 5 \times \left(\frac{7!}{5} + 1 \right) $

Since $7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$ is divisible by $5$, the term inside the parentheses is an integer greater than 1.

Thus, the expression is composite.

Final Answer:

Both given expressions are composite numbers.

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