Today’s Puzzle:
It’s almost Halloween. I hope you enjoy this grave-marker puzzle. Write the numbers from 1 to 12 in both the first column and the top row so that those numbers and the given clues make a multiplication table.
Here’s the same puzzle, but it won’t use up all your printer ink.
Factors of 1683:
- 1683 is a composite number.
- Prime factorization: 1683 = 3 × 3 × 11 × 17, which can be written 1683 = 3² × 11 × 17.
- 1683 has at least one exponent greater than 1 in its prime factorization so √1683 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1683 = (√9)(√187) = 3√187.
- The exponents in the prime factorization are 2, 1, and 1. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 = 12. Therefore 1683 has exactly 12 factors.
- The factors of 1683 are outlined with their factor pair partners in the graphic below.
More About the Number 1683:
1683 is the hypotenuse of a Pythagorean triple:
792-1485-1683, which is (8-15-17) times 99.
1683 is the difference of two squares in SIX different ways:
842² – 841² = 1683,
282² – 279² = 1683,
98² – 89² = 1683,
82² – 71² = 1683,
58² – 41² = 1683, and
42² – 9² = 1683.
That last one means we are 81 numbers away from the next perfect square. I also highlighted a cool-looking difference.
1680, 1681, 1682, 1683, and 1684 are the second smallest set of FIVE consecutive numbers whose square roots can be simplified.
1683 is the odd leg of pythagorean triples. You have posted the m and n for the 6 triangles when you gave the difference of squares. The first pair 842 and 841 gives m minus n equals 1 and the ppt formed has a ratio of area to perimeter of 420.5(841÷2).
Yes, 1683 is a side in 8 Pythagorean triples, the two I listed and the 6 you mentioned. I didn’t even think of the ratio of the area to the perimeter. Thank you very much for sharing! Did you notice my 1681st post? I quoted you a lot.