780

Number
$780$ (seven hundred and eighty) is:


 * $2^2 \times 3 \times 5 \times 13$


 * The side length of the smallest square with integer sides that can be inscribed within two different Pythagorean triangles so that one side of the square lies on the hypotenuse:
 * $\tuple {1443, 1924, 2405}$ and $\tuple {1145, 2748, 2977}$ are the side lengths.


 * The first of the $4$th pair of triangular numbers whose sum and difference are also both triangular:
 * $780 = T_{39}$, $990 = T_{44}$, $780 + 990 = T_{59}$, $990 - 780 = T_{20}$


 * The $20$th hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$, $\ldots$, $378$, $435$, $496$, $561$, $630$, $703$:
 * $780 = \displaystyle \sum_{k \mathop = 1}^{20} \paren {4 k - 3} = 20 \paren {2 \times 20 - 1}$


 * The $39$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $325$, $351$, $378$, $406$, $435$, $465$, $496$, $528$, $561$, $595$, $630$, $666$, $703$, $741$:
 * $780 = \displaystyle \sum_{k \mathop = 1}^{39} k = \dfrac {39 \times \paren {39 + 1} } 2$

Also see

 * Smallest Square Inscribed in Two Pythagorean Triangles