Today’s Puzzle:
Every year I make some heart-shaped puzzles, but this heart is different: I haven’t used this design before. Can it win you over? Some of the clues are tricky, so make sure you use logic to find the one and only solution.
Factors of 1777:
- 1777 is a prime number.
- Prime factorization: 1777 is prime.
- 1777 has no exponents greater than 1 in its prime factorization, so √1777 cannot be simplified.
- The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1777 has exactly 2 factors.
- The factors of 1777 are outlined with their factor pair partners in the graphic below.
How do we know that 1777 is a prime number? If 1777 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1777. Since 1777 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, or 41, we know that 1777 is a prime number.
More About the Number 1777:
1777 is the sum of two squares:
39² + 16² = 1777.
1777 is the hypotenuse of a primitive Pythagorean triple:
1248-1265-1777, calculated from 2(39)(16), 39² – 16², 39² + 16²
Here’s another way we know that 1777 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 39² + 16² = 1777 with 39 and 16 having no common prime factors, 1777 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1777. Since 1777 is not divisible by 5, 13, 17, 29, 37, or 41, we know that 1777 is a prime number.
1777 looks interesting in some other bases:
It’s 12121 in base 6 because 1(6⁴) + 2(6³) + 1(6²) + 2(6¹) + 1(6º) = 1777,
2L2 in base 25, because 2(25²) + 21(25) + 2(1) = 1777, and
1B1in base 37, because 1(37²) + 11(37) + 1(1) = 1777.