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

Sol. We have, 7 x 11 x 13 + 13 = 1001 + 13 = 1014

1014 = 2 x 3 x 13 * 13

So, it is the product of more than two prime numbers. 2, 3 and 13.

Hence, it is a composite number. 7 x 6x 5x 4 x 3 x 2 x 1 x 3 x 2 x 1 + 5 = 5040+5=5045

5045 = 5 x 1009

It is the product of prime factors 5 and 1009.

Hence, it is a composite number.

11∗13∗15∗17+17=17 (11∗13∗15∗1+1)

11

∗

13

∗

15

∗

17

+

17

=

17

(

11

∗

13

∗

15

∗

1

+

1

)

Now, 11∗13∗15∗1

11

∗

13

∗

15

∗

1

is a product of odd numbers which is always odd.

Thus, we have 17∗ (odd+1)

17

∗

(

o

d

d

+

1

)

=odd∗ (odd+odd)

=

o

d

d

∗

(

o

d

d

+

o

d

d

)

=odd∗even

=

o

d

d

∗

e

v

e

n

=even

=

e

v

e

n

Now, 2 is the only prime number that is even. Hence, given number cannot be prime and is therefore composite