def compute():
# Consider an arbitrary fraction n/d:
# Let n = 10 * n1 + n0 be the numerator.
# Let d = 10 * d1 + d0 be the denominator.
# As stated in the problem, we need 10 <= n < d < 100.
# We must disregard trivial simplifications where n0 = d0 = 0.
#
# Now, a simplification with n0 = d0 is impossible because:
# n1 / d1 = n / d = (10*n1 + n0) / (10*d1 + n0).
# n1 * (10*d1 + n0) = d1 * (10*n1 + n0).
# 10*n1*d1 + n1*n0 = 10*d1*n1 + d1*n0.
# n1*n0 = d1*n0.
# n1 = d1.
# This implies n = d, which contradicts the fact that n < d.
# Similarly, we cannot have a simplification with n1 = d1 for the same reason.
#
# Therefore we only need to consider the cases where n0 = d1 or n1 = d0.
# In the first case, check that n1/d0 = n/d;
# in the second case, check that n0/d1 = n/d.
numer = 1
denom = 1
for d in range(10, 100):
for n in range(10, d):
n0 = n % 10
n1 = n // 10
d0 = d % 10
d1 = d // 10
if (n1 == d0 and n0 * d == n * d1) or (n0 == d1 and n1 * d == n * d0):
numer *= n
denom *= d
return str(denom // eulerlib.gcd(numer, denom))
def compute():
# Consider an arbitrary fraction n/d:
# Let n = 10 * n1 + n0 be the numerator.
# Let d = 10 * d1 + d0 be the denominator.
# As stated in the problem, we need 10 <= n < d < 100.
# We must disregard trivial simplifications where n0 = d0 = 0.
#
# Now, a simplification with n0 = d0 is impossible because:
# n1 / d1 = n / d = (10*n1 + n0) / (10*d1 + n0).
# n1 * (10*d1 + n0) = d1 * (10*n1 + n0).
# 10*n1*d1 + n1*n0 = 10*d1*n1 + d1*n0.
# n1*n0 = d1*n0.
# n1 = d1.
# This implies n = d, which contradicts the fact that n < d.
# Similarly, we cannot have a simplification with n1 = d1 for the same reason.
#
# Therefore we only need to consider the cases where n0 = d1 or n1 = d0.
# In the first case, check that n1/d0 = n/d;
# in the second case, check that n0/d1 = n/d.
numer = 1
denom = 1
for d in range(10, 100):
for n in range(10, d):
n0 = n % 10
n1 = n // 10
d0 = d % 10
d1 = d // 10
if (n1 == d0 and n0 * d == n * d1) or (n0 == d1
and n1 * d == n * d0):
numer *= n
denom *= d
return str(denom // eulerlib.gcd(numer, denom))
def is_prime(n, k):
"""
Test if a number n is prime k-times.
:param n: The prime number to be tested.
:param k: The number of tests.
:return:
"""
if n <= 1 or n == 4:
return False
if n <= 3:
return True
if eulerlib.is_even(n):
return False
while k > 0:
# Take random int in [2, n-2]
a = random.randint(2, n-1)
# Check if a and n are co-prime.
if eulerlib.gcd(n, a) != 1:
return False
# Fermat's little theorem
if modpow(a, n-1, n) != 1:
return False
k -= 1
return True
def digit_cancelling_fractions():
lowest_nums = []
lowest_denoms = []
for den in range(10, 100):
if den % 10 == 0:
continue
for num in range(10, den):
num_digits = eulerlib.digits(num)
den_digits = eulerlib.digits(den)
if num % 10 == 0:
continue
digits_intersection = list(set(num_digits) & set(den_digits))
if len(digits_intersection) > 0:
num_digits.remove(digits_intersection[0])
den_digits.remove(digits_intersection[0])
num_lct = eulerlib.digits_to_int(num_digits)
den_lct = eulerlib.digits_to_int(den_digits)
cancelled_fraction = num_lct / den_lct
fraction = num / den
if fraction == cancelled_fraction:
lowest_nums.append(num_lct)
lowest_denoms.append(den_lct)
print('{} / {} = {} / {}'.format(num, den, num_lct, den_lct))
num_prod = eulerlib.product(lowest_nums)
den_prod = eulerlib.product(lowest_denoms)
gcd = eulerlib.gcd(num_prod, den_prod)
return den_prod / gcd
def count_all_unconcealed(prime): result = [] for e in range(prime - 1): if eulerlib.gcd(e, prime - 1) == 1: result.append(count_unconcealed(prime, e)) else: result.append(10**20) # Sentinel return result
def solve():
qtys = [0] * (bound+1)
for m in xrange(2, int(ceil((sqrt(1+2*bound) - 1)/2))+1):
for n in xrange(2 - (m+1)%2, m, 2):
if gcd(m,n) != 1: continue
perim = 2*m*(m+n)
if perim > bound: break
for x in xrange(perim, bound+1, perim):
qtys[x] += 1
return sum(1 for q in qtys if q == 1)
def solve():
for a in range(1, 10):
for b in range(1, 10):
ab = a*10 + b
for c in range(1, 10):
checkFrac(ab,b,c, a,c) # ab/bc = a/c
checkFrac(ab,c,b, a,c) # ab/cb = a/c
checkFrac(ab,a,c, b,c) # ab/ac = b/c
checkFrac(ab,c,a, b,c) # ab/ca = b/c
(nums, denoms) = zip(*fractions)
num = product(nums)
denom = product(denoms)
return denom // gcd(num, denom)
def find(n: int):
p = 1
q = 1
for a in range(1, n):
for b in range(1, n):
for c in range(1, n):
if f(a, b, c):
p *= (10 * a + b)
q *= (10 * b + c)
elif g(a, b, c):
p *= (10 * a + b)
q *= (10 * c + b)
r = lib.gcd(p, q)
p //= r
q //= r
return (p, q)
def getPnum(edgeMAX):
res = {}
m = 2
flag = True
while flag:
for n in range(m - 1, 0, -2):
if m * m + n * n <= edgeMAX:
if gcd(m, n) == 1:
if m * m + n * n in res:
res[m * m + n * n].append((2 * m * n, m * m - n * n))
else:
res[m * m + n * n] = [(2 * m * n, m * m - n * n)]
else:
flag = False
break
m += 1
return res
def getPnumAll(edgeMAX):
res = {}
m = 2
flag = True
while flag:
for n in range(m - 1, 0, -2):
if m * m + n * n <= edgeMAX:
if gcd(m, n) == 1:
t = edgeMAX // (m * m + n * n)
for i in range(1, t + 1):
if (m * m + n * n) * i in res:
res[(m * m + n * n) * i].append(
(2 * m * n * i, (m * m - n * n) * i))
else:
res[(m * m + n * n) * i] = [(2 * m * n * i,
(m * m - n * n) * i)]
else:
flag = False
break
m += 1
return res
def simplify(self):
c = eulerlib.gcd(self.num, self.den)
return Fraction(self.num // c, self.den // c)
def compute():
ans = 1
for i in range(1, 21):
ans *= i // eulerlib.gcd(i, ans)
return str(ans)
from eulerlib import cmpf, gcd N = 12000 f1 = (1,3) f2 = (1,2) c = 0 for d in xrange(1,N+1): for n in xrange(d/3,d/2+1): if gcd(n,d) == 1: f = (n,d) if cmpf(f,f1) > 0 and cmpf(f,f2) < 0: c += 1 print c
def totient(n):
result = 0
for k in xrange(1, n + 1):
if gcd(k, n) == 1:
result += 1
return result
def compute():
ans = 1
for i in range(1, 21):
ans *= i // eulerlib.gcd(i, ans)
return str(ans)