560

The smallest tetrahedral number that is the product of two tetrahedral numbers both greater than one. Also the second smallest number and smallest tetrahedral number divisible by 1, 4, 10, 20, 35, and 56 (the first six tetrahedral numbers; the smallest number divisible by all those is 280)

tetra(3) * tetra(6) = (3*4*5/6)*(6*7*8/6) = (3*4*5/6)*(7*8) = (60/6)*56 = 10*56 = 560 = tetra(14). 560/4=140, 560/10= 56, 560/20= 28, 560/35= 16, 560/56= 10 (all quotients are even so 280, half of 560, is divisible by all divisors given (1, 4, 10, 20, 35, and 56)).

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Tetrahedral number

The {n}th tetrahedral number is the sum of the first n Triangular numbers. Triangular numbers are the sum of the first n integers (1, 3, 6, 10, 15, etc.), and taking the sum of the triangular numbers results in the tetrahedral numbers (1, 4, 10, 20, 35, etc.)

The fifth tetrahedral number is the sum of the first five triangular numbers. The first five triangular numbers are 1, 3=1+2, 6=1+2+3, 10=1+2+3+4, and 15=1+2+3+4+5, so the sum of these numbers is 1+3+6+10+15= 35.

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