# Euler problems/181 to 190

## 1 Problem 181

Investigating in how many ways objects of two different colours can be grouped.

## 2 Problem 182

RSA encryption.

Solution:

```fun a1 b1 = sum [ e | e <- [2..a*b-1],
gcd e (a*b) == 1,
gcd (e-1) a == 2,
gcd (e-1) b == 2 ]
where a = a1-1
b = b1-1

problem_182 = fun 1009 3643```

## 3 Problem 183

Maximum product of parts.

Solution:

```-- Does the decimal expansion of p/q terminate?
terminating p q = 1 == reduce [2,5] (q `div` gcd p q)
where reduce   []   n = n
reduce (x:xs) n | n `mod` x == 0 = reduce (x:xs) (n `div` x)
| otherwise      = reduce xs n

-- The expression (round \$ fromIntegral n / e) computes the integer k
-- for which (n/k)^k is at a maximum. Also note that, given a rational number
-- r and a natural number k, the decimal expansion of r^k terminates if
-- and only if the decimal expansion of r does.
answer = sum [if terminating n (round \$ fromIntegral n / e) then -n else n
| n <- [5 .. 10^4]]
where e = exp 1

`problem_184 = undefined`