1699 Sweet Candy Cane

Today’s Puzzle:

Solving this candy cane puzzle can be a sweet experience. Just use logic to write the numbers 1 to 12 in both the first column and the top row so that those numbers and the given clues make a multiplication table.

Factors of 1699:

  • 1699 is a prime number.
  • Prime factorization: 1699 is prime.
  • 1699 has no exponents greater than 1 in its prime factorization, so √1699 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1699 has exactly 2 factors.
  • The factors of 1699 are outlined with their factor pair partners in the graphic below.

How do we know that 1699 is a prime number? If 1699 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1699. Since 1699 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, or 41, we know that 1699 is a prime number.

More About the Number 1699:

1699 is the third prime in a prime triple. What are the other two primes in that triple?

1699 is the difference of two squares:
850² – 849² = 1699.

1493 and Level 4

Today’s Puzzle:

Write the numbers from 1 to 12 in both the first column and the top row so that those numbers are the factors of the clues given in the puzzle:

Factors of 1493:

  • 1493 is a prime number.
  • Prime factorization: 1493 is prime.
  • 1493 has no exponents greater than 1 in its prime factorization, so √1493 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1493 has exactly 2 factors.
  • The factors of 1493 are outlined with their factor pair partners in the graphic below.

How do we know that 1493 is a prime number? If 1493 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1493. Since 1493 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1493 is a prime number.

Other Facts about the number 1493:

1493 is the last prime number in the fourth prime quintuplet,
(1481, 1483, 1487, 1489, 1493), which is the smallest prime quintuplet that is not also part of a prime sextuplet.
In prime quintuplets, the first three numbers, the middle three numbers, and the last three numbers each form a prime triplet. Thus,1493 is the last prime number in the third prime triplet formed from the numbers in the fourth prime quintuplet.

1493 is the sum of two squares:
38² + 7² = 1493

1493 is the hypotenuse of a Pythagorean triple:
532-1395-1493, calculated from 2(38)(7), 38² – 7², 38² + 7²

Here’s another way we know that 1493 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 38² + 7² = 1493 with 38 and 7 having no common prime factors, 1493 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1493. Since 1493 is not divisible by 5, 13, 17, 29, or 37, we know that 1493 is a prime number.