### 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.