1313 Virgács and St. Nickolas Day

6 December is Saint Nickolas Day. Children in Hungary and other places in Europe wake up to find candy and virgács in their boots. You can read more about this wonderful tradition in Jön a Mikulás (Santa is Coming) or Die Feier des Weihnachtsmanns (The Celebration of Santa Claus). Today’s puzzle represents the virgács given to children who have been even the least bit naughty during the current year.

Print the puzzles or type the solution in this excel file: 12 factors 1311-1319

Now I’ll write a little bit about the number 1313:

  • 1313 is a composite number.
  • Prime factorization: 1313 = 13 × 101
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1313 has exactly 4 factors.
  • Factors of 1313: 1, 13, 101, 1313
  • Factor pairs: 1313 = 1 × 1313 or 13 × 101
  • 1313 has no square factors that allow its square root to be simplified. √1313 ≈ 36.23534

1313 is the sum of consecutive prime numbers in three different ways:
It is the sum of the twenty-one prime numbers from 19 to 107.
It is the sum of eleven consecutive primes:
97 + 101 + 103 + 107 + 109 + 113 + 127 + 131 + 137 + 139 + 149 = 1313,
and it is the sum of seven consecutive prime numbers:
173 + 179 + 181 + 191 + 193 + 197 + 199 = 1313

1313 is the sum of two squares two different ways:
32² + 17² = 1313
28² +  23² = 1313

1313 is the hypotenuse of FOUR Pythagorean triples:
255-1288-1313 calculated from 28² –  23², 2(28)(23), 28² +  23²
260-1287-1313 which is 13 times (20-99-101)
505-1212-1313 which is (5-12-13) times 101
735-1088-1313 calculated from 32² – 17², 2(32)(17), 32² + 17²

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