Problem:
Score : 500 points

Problem Statement

For integers b (b \geq 2) and n (n \geq 1), let the function f(b,n) be defined as follows:

f(b,n) = n, when n < b

f(b,n) = f(b,\,{\rm floor}(n / b)) + (n \ {\rm mod} \ b), when n \geq b

Here, {\rm floor}(n / b) denotes the largest integer not exceeding n / b,
and n \ {\rm mod} \ b denotes the remainder of n divided by b.

Less formally, f(b,n) is equal to the sum of the digits of n written in base b.
For example, the following hold:

f(10,\,87654)=8+7+6+5+4=30

f(100,\,87654)=8+76+54=138

You are given integers n and s.
Determine if there exists an integer b (b \geq 2) such that f(b,n)=s.
If the answer is positive, also find the smallest such b.

Constraints

1 \leq n \leq 10^{11}

1 \leq s \leq 10^{11}

n,\,s are integers.

Input

The input is given from Standard Input in the following format:

n
s

Output

If there exists an integer b (b \geq 2) such that f(b,n)=s, print the smallest such b.
If such b does not exist, print -1 instead.

Sample Input 1

87654
30

Sample Output 1

10

Sample Input 2

87654
138

Sample Output 2

100

Sample Input 3

87654
45678

Sample Output 3

-1

Sample Input 4

31415926535
1

Sample Output 4

31415926535

Sample Input 5

1
31415926535

Sample Output 5

-1

Solution in Ruby (2.3.3) (judge verdict: Accepted)

n = gets.to_i
s = gets.to_i

if s > n
  puts -1
  exit
elsif s == n
  puts n + 1
  exit
end

def f(b, n)
  return 0 if n == 0
  return f(b, n / b) + (n % b)
end

ans = 1e12

(2..(n ** 0.5)).each do |b|
  ans = b if s == f(b, n) && b < ans
end

(1..(n ** 0.5)).each do |a|
  if (n - s) % a == 0 && (n - s) / a != 0
    b = (n - s) / a + 1
    ans = b if s == f(b, n) && b < ans
  end
end

p ans == 1e12 ? -1 : ans