Divisibility rules

Learning Objectives Recall and apply basic divisibility rules for 2, 3, 4, 5, 6, 8, 9, and 10. Apply advanced divisibility rules for 7, 11, and 13 to determine factors of large integers. Formulate and apply divisibility tests for composite numbers by using their co-prime factors. Represent any integer algebraically as a sum of the products of its digits and powers of 10. Construct algebraic proofs for the divisibility rules of 3, 9, and 11. Solve number theory problems that involve finding unknown digits using divisibility properties. How can you tell if a massive number like 4,195,847,292 is divisible by 12 without touching a calculator? 🤯 Let's unlock the mathematical secrets to do just that! This tutorial moves beyond simple tricks to a deeper understanding of numb...

TermDefinitionExample DivisibilityAn integer 'a' is divisible by a non-zero integer 'b' if there exists an integer 'c' such that a = bc. This means the division a/b results in an integer with a remainder of zero.24 is divisible by 8 because 24 = 8 × 3. Co-prime NumbersTwo integers are co-prime (or relatively prime) if their greatest common divisor (GCD) is 1. They share no common factors other than 1.8 and 15 are co-prime because the factors of 8 are {1, 2, 4, 8} and the factors of 15 are {1, 3, 5, 15}. Their only common factor is 1. Algebraic Representation of an IntegerExpressing a number as a polynomial in base 10, where the coefficients are the digits of the number.The 4-digit number 5283 can be written algebraically as 5(10^3) + 2(10^2) + 8(10^1) + 3(10^...

Algebraic Representation of an Integer An n-digit integer N with digits d_{n-1}d_{n-2}...d_1d_0 can be expressed as: N = \sum_{i=0}^{n-1} d_i \cdot 10^i = d_{n-1}10^{n-1} + ... + d_110^1 + d_010^0 This formula is the foundation for proving all divisibility rules. It deconstructs a number into its place value components, allowing for algebraic manipulation. Divisibility by a Composite Number Let C be a composite number with co-prime factors a and b (i.e., C = ab and gcd(a,b)=1). An integer N is divisible by C if and only if N is divisible by both a and b. To test for divisibility by 36, you must test for divisibility by 4 and 9 (since 4x9=36 and gcd(4,9)=1). Testing by 6 and 6 is insufficient. Divisibility by 11 (Alternating Sum) An integer N is divisible by 11 if the a...