Today’s Puzzle:
Write all the numbers from 1 to 12 in both the first column and the top row so that those numbers are the factors of the given clues.
Factors of 1730:
- 1730 is a composite number.
- Prime factorization: 1730 = 2 × 5 × 173.
- 1730 has no exponents greater than 1 in its prime factorization, so √1730 cannot be simplified.
- The exponents in the prime factorization are 1, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 1730 has exactly 8 factors.
- The factors of 1730 are outlined with their factor pair partners in the graphic below.
More about the number 1730:
1730 is the sum of two squares in two different ways:
41² + 7² = 1730, and
37² + 19² = 1730.
1730 is the hypotenuse of FOUR Pythagorean triples:
520 1650 1730 which is 10 times (52-165-173)
574 1632 1730 calculated from 2(41)( 7), 41² – 7², 41² + 7²,
1008 1406 1730 calculated from 37² – 19², 2(37)(19), 37² + 19², and
1038 1384 1730 which is 346 times (3-4-5).
Finally, OEIS.org informs us that 1730 is the sum of consecutive squares in two different ways. What are those two ways? I figured it out. Can you?
Here’s a hint: It is the sum of three consecutive squares as well as twelve consecutive squares. That means √(1730/3) rounded is included in one sum and √(1730/12) rounded is included in the other. The solution can be found in the comments. Have fun finding them yourself though!