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Today’s Puzzle:
Candy corn is a triangular piece of Halloween candy. 1684 is a centered triangular number formed from the sum of the 32nd, the 33rd, and the 34th triangular numbers. Label the boxes next to the representations of each of those triangular numbers.
Factors of 1684:
- 1684 is a composite number.
- Prime factorization: 1684 = 2 × 2 × 421, which can be written 1684 = 2² × 421.
- 1684 has at least one exponent greater than 1 in its prime factorization so √1684 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1684 = (√4)(√421) = 2√421.
- The exponents in the prime factorization are 2 and 1. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1) = 3 × 2 = 6. Therefore 1684 has exactly 6 factors.
- The factors of 1684 are outlined with their factor pair partners in the graphic below.
More About the Number 1684:
1684 is the sum of two squares:
30² + 28² = 1684.
1684 is the hypotenuse of a Pythagorean triple:
116-1680-1684, calculated from 30² – 28², 2(30)(28), 30² + 28².
It is also 4 times (29-420-421).
1680, 1681, 1682, 1683, and 1684 are the second smallest set of FIVE consecutive numbers whose square roots can be simplified.
1684/2 = 842, which is the third number in the smallest set of FIVE consecutive numbers whose square roots can be simplified.