def check(x,y):
if x > 2*y: return False
d = pe.gcd(x,y)
x/=d
y/=d
its = 0
while y > 1:
if x >= 2*y : return its%2
x -= y
its += 1
x,y = max(x,y), min(x,y)
# print x,y
return its%2
cap = 10**12
mod = 10**5
t = 1
while t <= cap:
t5 = 1
while t*t5 <= cap:
h.append(t*t5)
t5 *= 5
t *= 2
o2 = prime_fact_ord(2,cap)
o5 = prime_fact_ord(5,cap)
exp2 = o2-o5 #Significant exponent of 2 in final product
h = sorted(h) #numbers divisible only by 2 and 5
print len(h)
res = 1 #init to one
for i in xrange(1,min(cap+1,mod)):
if gcd(i,10) == 1:
e = 0 #the exponent by i in the final product
for m in h:
if m*i > cap:
break
max_r = ((cap/m) - i) / mod #maximum r with m*(r*mod+i) <= cap
e += max_r+1
res = (res * pow(i,e,mod)) % mod
res = (res * pow(2,exp2,mod)) % mod
print res
primes,mask = pe.primes_and_mask(cap)
curr = np.ones(cap,dtype=np.int64)
for croak in croaks:
prev = curr
curr = np.ones(cap,dtype=np.int64)
#Count croaks multiplicities. For each path on a prime, P gets croaked twice
#For each on a non-prime, N gets croaked twice
if croak == "P":
for i in xrange(cap):
if mask[i+1]:
prev[i] *= 2
else:
for i in xrange(cap):
if not mask[i+1]:
prev[i] *= 2
#compute transitions
curr[1:cap-1] = prev[:cap-2] + prev[2:cap]
curr[0] = prev[1]
curr[cap-1] = prev[cap-2]
#double count the contributions from the end squares
curr[1] += prev[0]
curr[cap-2] += prev[cap-1]
num = np.sum(prev)
print "Total admissible possibilities " + str(num)
d = pe.gcd(denom,num)
print "Reduced fraction " + str(num/d) + "/" + str(denom/d)