Explain why 7×11×13+13 and 7×6×5×4×3×2×1+5 are composite numbers.

What are Composite Numbers?

Composite numbers are positive integers greater than 1 that have factors other than 1 and themselves. In other words, a composite number is a number that can be divided evenly by at least one number other than 1 and itself. 

For example- 4 is a composite number because it is divisible by 2, in addition to 1 and 4. Similarly, numbers like 9, 12, 15, and 20 are also composite numbers since they have multiple factors. On the other hand, prime numbers, such as 2, 3, 5, and 7, are not composite numbers because they only have two distinct factors: 1 and the number itself.

Solution:

Let’s simplify the expressions and determine whether they are composite numbers.

For 7 × 11 × 13 + 13:

7 × 11 × 13 + 13 = 1001 + 13 = 1014

To check if 1014 is composite, let’s find its factors:

1014 can be factored as 2 × 3 × 13 × 13

Since 1014 has more than two factors (2, 3, 13, and 13), it is indeed a composite number.

Now let’s simplify 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5:

7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 = 5040 + 5 = 5045

To check if 5045 is composite, let’s find its factors:

5045 can be factored as 5 × 1009

Similarly, since 5045 has more than two factors (5 and 1009), it is also a composite number.

In conclusion, both 7 × 11 × 13 + 13 (1014) and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 (5045) are composite numbers.

Summary:

Explain why 7×11×13+13 and 7×6×5×4×3×2×1+5 are composite numbers.

The numbers 1014 and 5045, which are obtained by simplifying 7×11×13+13 and 7×6×5×4×3×2×1+5, respectively, are composite numbers. They possess factors beyond 1 and themselves. For instance, 1014 can be factorized into 2×3×13×13, while 5045 can be expressed as 5×1009. Since both numbers have more than two factors, they fall into the category of composite numbers.

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