# 1684 Triangular Candy Corn

Contents

### 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.

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