Question

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

Answer 1

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

Answer 2

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.