1714 Will the Factors in This Puzzle Give You Fits?

Today’s Puzzle:

12 and 24 have several common factors, but only one of them works in this puzzle. Will it be 2, 3, 4, 6, or 12?

What about 40 and 60’s common factors?

Don’t guess which factor to use. Start elsewhere in the puzzle where there’s only one possible common factor. Then use logic to eliminate some of the factor possibilities for 12, 24 and 40, 60. You will have to think, but it won’t be too difficult.

Factors of 1714:

  • 1714 is a composite number.
  • Prime factorization: 1714 = 2 × 857.
  • 1714 has no exponents greater than 1 in its prime factorization, so √1714 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 1714 has exactly 4 factors.
  • The factors of 1714 are outlined with their factor pair partners in the graphic below.

More About the Number 1714:

1714 is the sum of two squares:
33² + 25² = 1714.

1714 is the hypotenuse of a Pythagorean triple:
464-1650-1714, calculated from 33² – 25², 2(33)(25), 33² + 25².
It is also 2 times (232-825-857).

1712 Can You Make the Factors Fit?

Today’s Puzzle:

This Factor Fits puzzle starts off fairly easy before it potentially might give you fits trying to place the rest of the factors. Are you game?

Factors of 1712:

  • 1712 is a composite number.
  • Prime factorization: 1712 = 2 × 2 × 2 × 2 × 107, which can be written 1712 = 2⁴ × 107.
  • 1712 has at least one exponent greater than 1 in its prime factorization so √1712 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1712 = (√16)(√107) = 4√107.
  • The exponents in the prime factorization are 4 and 1. Adding one to each exponent and multiplying we get (4 + 1)(1 + 1) = 5 × 2 = 10. Therefore 1712 has exactly 10 factors.
  • The factors of 1712 are outlined with their factor pair partners in the graphic below.

More About the Number 1712:

1712 is the difference of two squares in three different ways:
429² – 427² = 1712,
216² – 212² = 1712, and
111² – 103² = 1712.

1708 Happy Birthday, Jo Morgan!

Today’s Puzzle:

A few days ago I published a new kind of factoring puzzle. Jo Morgan of Resourceaholic.com, keeps an eye out for new mathematical resources on Twitter. She was one of the first to notice and like my puzzle. Because of her, lots of other people noticed the puzzle, too. Today is Jo’s birthday, and I decided to make a similar puzzle for her to enjoy. You might find it slightly more difficult than the earlier puzzle, but use logic from the beginning, and you will be able to fit in all the factors.

Factors of 1708:

  • 1708 is a composite number.
  • Prime factorization: 1708 = 2 × 2 × 7 × 61, which can be written 1708 = 2² × 7 × 61.
  • 1708 has at least one exponent greater than 1 in its prime factorization so √1708 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1708 = (√4)(√427) = 2√427.
  • 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 1708 has exactly 12 factors.
  • The factors of 1708 are outlined with their factor pair partners in the graphic below.

More About the Number 1708:

1708 is the hypotenuse of a Pythagorean triple:
308-1680-1708, which is 28 times (11-60-61).