Divisibility rules for 2, 5, and 10

Learning Objectives Algebraically prove why the divisibility rules for 2, 5, and 10 work using polynomial representations of integers. Analyze a polynomial expression P(n) and determine the conditions on the integer variable 'n' that make P(n) divisible by 2, 5, or 10. Decompose variable expressions into components that are demonstrably divisible by 2, 5, or 10 and a remaining component to be analyzed. Construct variable expressions that are always divisible by 2, 5, or 10 for any integer input. Apply the concept of divisibility to solve problems involving sequences and series where terms must meet specific criteria. Evaluate the divisibility of complex algebraic expressions by identifying the controlling term(s). How can we predict if a complex algebraic formula,...

TermDefinitionExample DivisibilityAn integer 'a' is divisible by a non-zero integer 'b' if there exists an integer 'c' such that a = bc. In the context of expressions, a variable expression P(n) is divisible by 'b' if P(n) = b * Q(n) for some expression Q(n) that yields an integer for any integer 'n'.The expression 10n + 5 is divisible by 5 because it can be factored as 5(2n + 1). Since 'n' is an integer, '2n+1' is also an integer. Polynomial Representation of an IntegerAny integer can be expressed as a sum of powers of 10. This is the basis of our decimal system and is crucial for proving divisibility rules algebraically.The number 483 can be written as 4 * 10^2 + 8 * 10^1 + 3 * 10^0. We can generalize this as N = a_k...

Algebraic Rule for Divisibility by 2 An expression P(n) is divisible by 2 if P(n) \equiv 0 \pmod{2}. For any integer N = 10k + d_0, its divisibility by 2 is determined entirely by its last digit, d_0, because 10k is always divisible by 2. To check if a variable expression is divisible by 2, decompose it into terms that are always even (e.g., multiples of 2, 10, etc.) and analyze the remaining terms. The expression is divisible by 2 if and only if the sum of the remaining terms is divisible by 2. Algebraic Rule for Divisibility by 5 An expression P(n) is divisible by 5 if P(n) \equiv 0 \pmod{5}. For any integer N = 10k + d_0, its divisibility by 5 is determined entirely by its last digit, d_0, because 10k is always divisible by 5. To check if a variable expression is divisibl...