Test the following numbers for compositeness. If they are composite, give their prime decomposition:
(a) $(2 × 3 × 5 × 7) − 1$
(b) $(2 × 3 × 5 × 7) + 1$
(a) $(2 × 3 × 5 × 7) − 1 = 209$. We only need to test prime factors less than or equal to $\lfloor \sqrt{209} \rfloor = 14$. However, the form $(2 × 3 × 5 × 7) − 1$ tells us that 2,3,5 and 7 are not prime factors. So we only need to test 11 and 13.
This gives us
$ 209 = 11 \times 19 $
(b) $(2 × 3 × 5 × 7) + 1 = 211$. We only need to test prime factors less than or equal to $\lfloor \sqrt{211} \rfloor = 14$. However, the form $(2 × 3 × 5 × 7) + 1$ tells us that 2,3,5 and 7 are not prime factors. So we only need to test 11 and 13.
211 does not have any such factors, and so is prime.