from fractions import Fraction def xgcd(a, b): x0, y0, x1, y1 = 1, 0, 0, 1 while b != 0: q, a, b = a // b, b, a % b x0, x1 = x1, x0 - q * x1 y0, y1 = y1, y0 - q * y1 return a, x0, y0 def modinv(a, m): g, x, y = xgcd(a, m) if g != 1: raise Exception('Modular inverse does not exist') else: return x % m def solve1(p, q, sqrt): invp = modinv(p, q) # lower lower = Fraction(0, 1) def go1(x): nonlocal lower f = Fraction(int(p * x // q), x) lower = max(lower, f) for x in range(1, sqrt): # go1(x) go1(q - x) for y in range(1, sqrt): # px mod q = y go1(y * invp % q) return lower def solve2(p, q, sqrt): invp = modinv(p, q) # upper upper = Fraction(p, 1) def go2(x): nonlocal upper f = Fraction(int(p * x // q + 1), x) upper = min(upper, f) for x in range(1, sqrt): go2(x) # go2(q - x) for y in range(q - sqrt + 1, q): # px mod q = y go2(y * invp % q) return upper def main(): p, q = list(map(int, input().split())) sqrt = min(q, 5 * 10 ** 5) lower = solve1(p, q, sqrt) upper = solve2(p, q, sqrt) import sys print(upper, file=sys.stderr) print(lower, file=sys.stderr) print(upper.numerator + upper.denominator + lower.numerator + lower.denominator) def stress(): q = 100000 import numpy as np for _ in range(100): p = np.random.randint(1, q) import math if math.gcd(p, q) != 1: continue lower_b = solve1(p, q, q) lower = solve1(p, q, int(np.sqrt(q)) + 10) print(p, q, lower_b, lower) assert lower_b == lower # stress() main()