Today's question.. Thanks in advance!

yumi2012 wrote:Which of the following numbers is not prime? (Hint: avoid actually computing these numbers.)

A. 6!-1
B. 6!+21
C. 6!+41
D. 7!-1
E. 7!+11

---
D and E are not primes either.. Why can only B be the answer???

We can eliminate Option A and D by the below rule,

n!+1 is prime for n=1, 2, 3, 11, 27, 37, 41, 73, 77, 116, 154, 320, 340, 399, 427, 872, 1477, 6380, and 26951 (107707 digits).

n!-1 is prime for n=3, 4, 6, 7, 12, 14, 30, 32, 33, 38, 94, 166, 324, 379, 469, 546, 974, 1963, 3507, 3610, 6917, 21480, 34790, 94550, and 103040 (471794 digits).

But how to choose one from the remaining three options(B,C,E) ?


Regards,
Uva

we can factor out 3 out of 6!+21 which means that this number is not a prime

yumi2012 wrote:Which of the following numbers is not prime? (Hint: avoid actually computing these numbers.)

A. 6!-1
B. 6!+21
C. 6!+41
D. 7!-1
E. 7!+11

---
D and E are not primes either.. Why can only B be the answer???

Every prime number greater than 3 is of the form 6n+1 or 6n-1.

Let's look at the options:
A. 6! - 1 = 6n - 1
(A) can be a prime.

C. 6! + 41 = (6! + 42) - 1 = 6n - 1
(C) can be a prime.

D. 7! - 1 = 6n - 1
(D) can be a prime.

E. 7! + 11 = (7! + 12) - 1 = 6n - 1
(E) can be a prime.

B. 6! + 21 = 6! + 18 + 3 = 6n + 3
(B) is definitely not a prime.

Choose B

Cheers