Today’s Puzzle:

Is this mystery-level puzzle difficult or easy to solve? I’m not telling. You’ll have to try it for yourself to find out. As always, there is only one solution.

Factors of 1781:

1781 ÷ 4 leaves a remainder of 1, and 41² + 10² = 1781. Could 1781 be a prime number? It will be unless it has a prime number hypotenuse less than √1781 as a divisor. In other words, is it divisible by 5, 13, 17, 29, 37, or 41?

1781 obviously isn’t divisible by 5, and since it’s 41² + 10², it isn’t divisible by 41 either. That means we only have to check if it is divisible by 13, 17, 29, and 37.

So is it prime or composite?

• 1781 is a composite number.
• Prime factorization: 1781 = 13 × 137.
• 1781 has no exponents greater than 1 in its prime factorization, so √1781 cannot be simplified.
• The exponents in the prime factorization are 1 and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1781 has exactly 4 factors.
• The factors of 1781 are outlined with their factor pair partners in the graphic below.

More About the Number 1781:

Not only does 41² + 10² = 1781, but
34² + 25² = 1781.

That 34² lets us know right away that 1781 is not divisible by 17, but any number that is the sum of two squares in more than one way is never a prime number.

1781 is the hypotenuse of FOUR Pythagorean triples:

531-1700-1781, calculated from 34² – 25², 2(34)(25), 34² + 25²,
685-1644-1781, which is (5-12-13) times 137,
820-1581-1781, calculated from 2(41)(10), 41² – 10², 41² + 10², and
1144-1365-1781, which is 13 times (88-105-137).

1781 is also the difference of two squares in two different ways:
891² – 890² = 1781, and
75² – 62² = 1781.