# STEP II/2005/2

For any positive integer N, the function f(N) is defined by $f(N) = N\Big(1-\frac1{p_1}\Big)\Big(1-\frac1{p_2}\Big) \cdots\Big(1-\frac1{p_k}\Big)$

where $p_1, p_2, \dots , p_k$ are the only prime numbers that are factors of N.

Thus $f(80)=80(1-\frac{1}{2})(1-\frac{1}{5})\,$

(1) Evaluate f(12) and f(180)

(2) Show that f(N) is an integer for all N

(3) Prove, or disprove by means of a counterexample, each of the following:

(a) $f(m) f(n) = f(mn)$

(b) $f(p) f(q) = f(pq)$ if p and q are distinct prime numbers;

(c) $f(p) f(q) = f(pq)$ only if p and q are distinct prime numbers.

(4) Find a positive integer m and a prime number p such that $f(p^m) = 146410$

# STEP II/2018/6

(i) Find all pairs of positive integers (n,p), where p is a prime number, that satisfy $n!+ 5 = p$

(ii) In this part of the question you may use the following two theorems:

For $n\ge 7, 1! \times 3! \times \cdots \times (2n-1)! > (4n)!\,$

For every positive integer n, there is a prime number between 2n and 4n.

Find all pairs of positive integers (n,m) that satisfy $1! \times 3! \times \cdots \times (2n-1)! = m!$