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.
Question.

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.

Solution:

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

$=(77+1) \times 13$

$=78 \times 13=(2 \times 3 \times 13) \times 13$

So, $78=2 \times 3 \times 13$

$78 \times 13=2 \times 3 \times 13^{2}$

Since, $7 \times 11 \times 13+13$ can be expressed as a product of primes, therefore, it is a composite number.

(ii) $7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1+5$

$=(7 \times 6 \times 4 \times 3 \times 2 \times 1+1) \times 5$

$=1009 \times 5$

Since, $7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1+5$ can be expressed as a product of primes, therfore it is a composite number.
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