Proving divisibility results

Understand the definition of divisibility: One integer ‘a’ is said to divide another integer ‘b’ if and only if there exists an integer ‘c’ such that a*c=b.
Familiarity with Euclid’s algorithm is paramount. This procedure calculates the greatest common divisor (gcd) of two numbers and can help in establishing divisibility results.
Be able to prove basic divisibility tests like divisibility by 2, 3, 4, 5, 10 etc. For example, if the last digit of a number is 0, 2, 4, 6, or 8, then the number is divisible by 2.
Broadly grasp the concept of prime numbers. A prime number is only divisible by 1 and itself without a remainder.
Recognise that if for integers a, b, and c, ‘a’ divides ‘b’ and ‘b’ divides ‘c’, this implies that ‘a’ also divides ‘c’. This property will frequently be used in your solutions.
Understand the concept of congruence; ‘a’ is congruent to ‘b modulo m’, if ‘m’ divides (a-b). Congruences can simplify calculations dramatically.
Know the difference between commensurate quantities and incommensurate quantities. Commensurate quantities have a common measure, while incommensurate quantities do not.
Ensure you have a mastery of even and odd functions. An even function is symmetric about the y-axis (i.e. it is unchanged by x going to -x), hence the numbers are divisible by 2, whereas an odd function is symmetric about the origin (i.e. it changes sign when x goes to -x).
Remember, the sum of the digits of a number is divisible by 9 if and only if the number itself is divisible by 9. This rule can be incredibly useful for larger values.
Regularly practice mathematical proofs that follow from these principles, ensuring that each step has been justifiably derived and defined to reinforce understanding.