780

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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 $1$, $6$, $15$, $28$, $45$, $66$, $91$, $\ldots$, $378$, $435$, $496$, $561$, $630$, $703$:

$780 = \ds \sum_{k \mathop = 1}^{20} \paren {4 k - 3} = 20 \paren {2 \times 20 - 1}$

The $39$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $325$, $351$, $378$, $406$, $435$, $465$, $496$, $528$, $561$, $595$, $630$, $666$, $703$, $741$:

$780 = \ds \sum_{k \mathop = 1}^{39} k = \dfrac {39 \times \paren {39 + 1} } 2$

Also see

• Smallest Square Inscribed in Two Pythagorean Triangles

• Previous  ... Next: Hexagonal Number
• Previous  ... Next: Triangular Number Pairs with Triangular Sum and Difference
• Previous  ... Next: Triangular Number

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $780$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $780$