1713 A Little More of a Subtraction Distraction

Today’s Puzzle:

It occurred to me that as long as the last box is neither 1 nor 12 that I could leave the clue above it blank. Can you still solve the puzzle?

Factors of 1713:

  • 1713 is a composite number.
  • Prime factorization: 1713 = 3 × 571.
  • 1713 has no exponents greater than 1 in its prime factorization, so √1713 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 1713 has exactly 4 factors.
  • The factors of 1713 are outlined with their factor pair partners in the graphic below.

More About the Number 1713:

1713 is the difference of two squares in two different ways:
857² – 856² = 1713, and
287² – 284² = 1713.

1713 is the sum of two, three, and six consecutive numbers:
856 + 857 = 1713,
570 + 571 + 572 = 1713, and
283 + 284 + 285 + 286 + 287 + 288 = 1713.

Do you see any relationship between those two facts?

1709 Sometimes “Guess and Check” Is a Good Strategy

Today’s Puzzle:

Most of the puzzles I publish are logic puzzles, and I encourage you to find the logic of the puzzle and not guess and check.  However, guess and check is a legitimate strategy in mathematics, and it is a legitimate strategy to solve this particular puzzle.

Since one of the clues is -9, we know that the two boxes under it must be [1, 10], [2, 11], or [3, 12].

Suppose you assume it’s 1 – 10 = -9. If you fill out the rest of the boxes you would get:

You know that isn’t right because zero is not a number from 1 to 12. No problem. Simply add one to each of the numbers you wrote in, and the puzzle will be solved with only numbers from 1 to 12.

Suppose you assumed it’s 3 -12 = -9. The rest of the boxes would look like this:

Again, 13 is not included in the numbers from 1 to 12, but you can fix it by subtracting 1 from each of the numbers you wrote in. Easy Peasy.

Factors of 1709:

  • 1709 is a prime number.
  • Prime factorization: 1709 is prime.
  • 1709 has no exponents greater than 1 in its prime factorization, so √1709 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1709 has exactly 2 factors.
  • The factors of 1709 are outlined with their factor pair partners in the graphic below.

How do we know that 1709 is a prime number? If 1709 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1709. Since 1709 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, or 41, we know that 1709 is a prime number.

More About the Number 1709:

1709 is the sum of two squares:
35² + 22² = 1709.

1709 is the hypotenuse of a Pythagorean triple:
741-1540-1709, calculated from 35² – 22², 2(35)(22), 35² + 22².

Here’s another way we know that 1709 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 35² + 22² = 1709 with 35 and 22 having no common prime factors, 1709 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1709. Since 1709 is not divisible by 5, 13, 17, 29, 37, or 41, we know that 1709 is a prime number.

1707 Subtraction Distraction

Today’s Puzzle:

Even though this Subtraction Distraction puzzle has more boxes than the one I published a couple of weeks ago, it is actually an easier puzzle. Can you write the numbers 1 to 12 in the boxes so that each triangle is its adjacent left box minus its adjacent right box?

Factors of 1707:

1 + 7 + 0 + 7 = 15, a multiple of 3, so 1707 is divisible by 3.

  • 1707 is a composite number.
  • Prime factorization: 1707 = 3 × 569.
  • 1707 has no exponents greater than 1 in its prime factorization, so √1707 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 1707 has exactly 4 factors.
  • The factors of 1707 are outlined with their factor pair partners in the graphic below.

More About the Number 1707:

1707 is the hypotenuse of a Pythagorean triple:
693-1560-1707, which is 3 times (231-520-569).