18. For some positive integer $n$, the prime factorisation of the number $n!=1 \times 2 \times \ldots \times n$ is of the form shown below.

The primes are written in increasing order. Ink has covered some of the primes and some of the exponents. What is the exponent of 17 ?

A 1
B 2
C 3
D 4
E 5
Solution
$\mathbf{C}$

The appearance of the prime 47 shows that $47 \leq n<53$ (otherwise we would see the next prime 53 ).
Since $13^{4}$ is part of the factorisation, 4 multiples of 13 are included in $n$ !. These are 13, 26, 39 and 52 , so $n=52$. There are three multiples of 17 before 52 (these are 17, 34 and 51). Therefore, the exponent of 17 in this factorisation is 3 .