def mystery(x):
if not isinstance(x, int) or x <= 0:
return False
n, p = x, []
d = 2
while d * d <= n:
while n % d == 0:
p.append(d)
n //= d
d += 1
if n > 1:
p.append(n)
s = sum(i * j for i, j in enumerate(sorted(p)))
return len(p) == 4 and len(set(p)) == 4 and s == 174 and x % 997 == 565
def mystery(x):
if not isinstance(x, int) or x <= 0:
return False
# perfect square?
if int(x ** 0.5) ** 2 != x:
return False
# sum of digits prime?
s = sum(int(d) for d in str(x))
if s < 2:
return False
for i in range(2, int(s ** 0.5) + 1):
if s % i == 0:
return False
# reverse prime?
rx = int(str(x)[::-1])
if rx < 2:
return False
for i in range(2, int(rx ** 0.5) + 1):
if rx % i == 0:
return False
# exactly three 1's in binary?
if bin(x).count('1') != 3:
return False
# modulo condition
if x % 123 != 46:
return False
return True
def mystery(x):
if not isinstance(x, int) or x < 0:
return False
b = bin(x)[2:]
# Exactly 20 bits
if len(b) != 20:
return False
# Exactly 10 ones
if b.count('1') != 10:
return False
# Binary palindrome
if b != b[::-1]:
return False
# Decimal digit sum constraint
digit_sum = sum(int(d) for d in str(x))
if digit_sum != 24:
return False
# Specific modular constraint
if x % 9973 != 2067:
return False
return True
def mystery(x):
if not isinstance(x, int) or x <= 0:
return False
# Modulo condition: last three digits 729
if x % 1000 != 729:
return False
# Harshad number: divisible by sum of digits
digit_sum = sum(int(d) for d in str(x))
if digit_sum == 0 or x % digit_sum != 0:
return False
# Binary weight: exactly 6 ones
if bin(x).count('1') != 6:
return False
# Check sum of two cubes in two distinct ways
max_a = int(round(x ** (1/3))) + 1
pairs = []
for a in range(1, max_a):
a3 = a * a * a
if a3 > x:
break
for b in range(a, max_a):
b3 = b * b * b
s = a3 + b3
if s > x:
break
if s == x:
pairs.append((a, b))
if len(pairs) >= 2:
break
if len(pairs) >= 2:
break
if len(pairs) < 2:
return False
# Check Carmichael number using Korselt's criterion
# Factor x by trial division
n = x
factors = []
d = 2
while d * d <= n:
while n % d == 0:
factors.append(d)
n //= d
d += 1 if d == 2 else 2
if n > 1:
factors.append(n)
# Must be composite
if len(factors) == 1:
return False
# Must be squarefree
if len(set(factors)) != len(factors):
return False
# For each prime factor p, p-1 divides x-1
for p in set(factors):
if (x - 1) % (p - 1) != 0:
return False
return True
def mystery(x):
if not isinstance(x, int) or x <= 0:
return False
s = str(x)
if len(s) != 8:
return False
digits = [int(d) for d in s]
if 0 in digits:
return False
first_half = int(s[:4])
r1 = int(first_half ** 0.5 + 0.5)
if r1 * r1 != first_half:
return False
last_half = int(s[4:])
r2 = int(last_half ** 0.5 + 0.5)
if r2 * r2 != last_half:
return False
if r1 + r2 != 117 or r1 * r2 != 2916:
return False
if x % 97 != 86:
return False
if digits[3] != digits[4]:
return False
return True
def mystery(x):
if not isinstance(x, int) or x <= 0:
return False
s = str(x)
# 7-digit palindrome
if len(s) != 7:
return False
if s != s[::-1]:
return False
# exactly 4 distinct digits
if len(set(s)) != 4:
return False
# perfect square
r = int(x ** 0.5)
if r * r != x:
return False
# square root is also a palindrome
rs = str(r)
if rs != rs[::-1]:
return False
# sum of digits of x is a perfect square
sx = sum(int(d) for d in s)
if int(sx ** 0.5) ** 2 != sx:
return False
# sum of digits of square root is a perfect square
sr = sum(int(d) for d in rs)
if int(sr ** 0.5) ** 2 != sr:
return False
# binary weight exactly 10
if bin(x).count('1') != 10:
return False
# specific modular condition
if x % 997 != 35:
return False
return True
def mystery(x):
if not isinstance(x, int) or x <= 0:
return False
n = x
factors = []
d = 2
while d * d <= n:
while n % d == 0:
factors.append(d)
n //= d
d += 1
if n > 1:
factors.append(n)
if len(factors) != 6:
return False
if len(set(factors)) != 3:
return False
from collections import Counter
counts = Counter(factors)
if not all(c == 2 for c in counts.values()):
return False
if sum(set(factors)) != 100:
return False
if x % 9973 != 2426:
return False
return True
def mystery(x):
if not isinstance(x, int) or x <= 0:
return False
# 8 digits
if x < 10_000_000 or x > 99_999_999:
return False
s = str(x)
if '0' in s:
return False
# Decompose into YYYYMMDD
year = int(s[:4])
month = int(s[4:6])
day = int(s[6:8])
# Specific month and day
if month != 11 or day != 13:
return False
# Year must be prime
if year < 2:
return False
for i in range(2, int(year**0.5) + 1):
if year % i == 0:
return False
# Sum of digits of year must be prime
year_digit_sum = sum(int(d) for d in str(year))
if year_digit_sum < 2:
return False
for i in range(2, int(year_digit_sum**0.5) + 1):
if year_digit_sum % i == 0:
return False
# Check that the date is a Friday using Zeller's congruence
# weekday: 0=Saturday, 1=Sunday, ..., 6=Friday
def weekday(y, m, d):
if m < 3:
m += 12
y -= 1
K = y % 100
J = y // 100
h = (d + (13*(m+1))//5 + K + K//4 + J//4 - 2*J) % 7
return h
if weekday(year, month, day) != 6:
return False
# Product of all digits must be a perfect cube
prod = 1
for ch in s:
prod *= int(ch)
# perfect cube check
root = round(prod ** (1/3))
if root**3 != prod:
return False
# Specific modular condition
if x % 997 != 374:
return False
return True
def mystery(x):
if not isinstance(x, int) or x <= 0:
return False
s = str(x)
if len(s) != 12:
return False
parts = [int(s[i:i+3]) for i in range(0, 12, 3)]
cubes = [1, 8, 27, 64, 125, 216, 343, 512, 729]
if any(p not in cubes for p in parts):
return False
cube_to_root = {1:1, 8:2, 27:3, 64:4, 125:5, 216:6, 343:7, 512:8, 729:9}
digits = [cube_to_root[p] for p in parts]
if len(set(digits)) != 4:
return False
if sum(digits) != 20:
return False
prod = 1
for d in digits:
prod *= d
root = int(prod ** 0.5 + 0.5)
if root * root != prod:
return False
if x % 9973 != 3498:
return False
return True
def mystery(x):
if not isinstance(x, int) or x <= 0:
return False
s = str(x)
if s != s[::-1]:
return False
# perfect square
r = int(x ** 0.5 + 0.5)
if r * r != x:
return False
# square root palindrome
rs = str(r)
if rs != rs[::-1]:
return False
# sum of digits perfect square
sd = sum(int(d) for d in s)
if int(sd ** 0.5 + 0.5) ** 2 != sd:
return False
# product of digits perfect square
prod = 1
for d in s:
prod *= int(d)
if int(prod ** 0.5 + 0.5) ** 2 != prod:
return False
# binary weight exactly 10
if bin(x).count('1') != 10:
return False
# modular condition
if x % 997 != 35:
return False
return True