Divisibility Rules and Polygon Combinatorics

Flashcards covering divisibility rules for integers and combinatorial formulas for triangles and diagonals formed by the vertices of an n-sided polygon.

Divisibility Rule for 2

A number is divisible by 2 if its last digit is an even number.

Divisibility Rule for 3

A number is divisible by 3 if the sum of its digits is divisible by 3.

Divisibility Rule for 4

A number is divisible by 4 if its last two digits form a number divisible by 4.

Divisibility Rule for 5

A number is divisible by 5 if the last digit of the number is 0 or 5.

Divisibility Rule for 6

A number is divisible by 6 if the number is divisible by both 2 and 3.

Divisibility Rule for 7

Double the units place digit and subtract it from the rest of the number to check if the result is divisible by 7.

Divisibility Rule for 8

A number is divisible by 8 if its last three digits form a number divisible by 8.

Divisibility Rule for 9

A number is divisible by 9 if the sum of its digits is divisible by 9.

Divisibility Rule for 10

A number is divisible by 10 if the last digit of the number is 0.

Divisibility Rule for 11

A number is divisible by 11 if the difference between the sum of digits at even places and odd places is divisible by 11.

Divisibility Rule for 25

A number is divisible by 25 if its last two digits are divisible by 25 (such as 00, 25, 50, or 75).

Number of Diagonals in an $n$-Sided Polygon

The total number of diagonals formed in a polygon with $n$ sides (or vertices), given by $\binom{n}{2} - n$.

Triangles Having One Side Common with an $n$-Sided Polygon

The number of triangles formed by joining vertices of an $n$-sided polygon that share exactly one side with the polygon, given by $n(n - 4)$.

Triangles Having Two Sides Common with an $n$-Sided Polygon

The number of triangles formed by joining vertices of an $n$-sided polygon that share two sides with the polygon, given by $n$.

Triangles Having No Side Common with an $n$-Sided Polygon

The number of triangles formed by joining vertices of an $n$-sided polygon that share no sides with the polygon, given by $\binom{n}{3} - (n + n(n - 4))$.

Total Triangles Formed by Vertices of an $n$-Sided Polygon ($P_n$)

The total number of triangles formed by selecting any 3 vertices of an $n$-sided regular polygon, given by $P_n = \binom{n}{3}$.