16. Andrew states that every composite number of the form $8 n+3$, where $n$ is an integer, has a prime factor of the same form.
Which of these numbers is an example showing that Andrew's statement is false?
A 19
B 33
C 85
D 91
E 99
SOLUTION
D To show that Andrew's statement is wrong we need to find a composite number of the form $8 n+3$ which does not have any prime factors of this form. This rules out 19 which is not composite, and both 33 and 85 which are not of the form $8 n+3$. The number 99 is of the form $8 n+3$, but it has two prime factors 3 and 11 which are also of this form. So 99 won't do either. This leaves 91 . We see that $91=8 \times 11+3$, and so it is of the form $8 n+3$, but $91=7 \times 13$ and neither of its prime factors, 7 and 13 , is of this form. So the number 91 shows that Andrew's statement is false.