Star of David theorem

The Star of David theorem is a mathematical result on arithmetic properties of binomial coefficients.

The greatest common divisors of the binomial coefficients forming each of the two triangles in the Star of David shape in Pascal's triangle are equal: Rows 8, 9, and 10 of Pascal's triangle are For n=9, k=3 or n=9, k=6, the element 84 (circled bold) is surrounded by, in sequence, the elements 28, 56, 126, 210, 120 and 36 (bold).

The element 36 (circled italics) is surrounded by the sequence 8, 28, 84, 120, 45 and 9 (italics), and taking alternating values we have gcd(8, 84, 45) = 1 = gcd(28, 120, 9).

The above greatest common divisor also equals

[1] Thus in the above example for the element 84 (in its rightmost appearance), we also have gcd(70, 56, 28, 8) = 2.

The two sets of three numbers which the Star of David theorem says have equal greatest common divisors also have equal products.

This result can be confirmed by writing out each binomial coefficient in factorial form, using