Divisibility Rules & Remainder Theorem

Understanding the relationship between numbers is key to solving divisibility and remainder problems.

Core Concept: The fundamental division equation is Dividend = (Divisor × Quotient) + Remainder. The remainder must always be non-negative and smaller than the divisor.
Factors & Multiples: If a number 'a' divides 'b' exactly (remainder is 0), then 'a' is a factor of 'b', and 'b' is a multiple of 'a'.

These rules and theorems are the most powerful tools for this topic.

1. Divisibility Rules at a Glance:

By 3 or 9: If the sum of the digits is divisible by 3 or 9 respectively.
By 4 or 8: If the number formed by the last two (for 4) or last three (for 8) digits is divisible by 4 or 8 respectively.
By 6: If the number is divisible by both 2 (is even) and 3.
By 7: Double the last digit and subtract it from the rest of the number. If the result is 0 or divisible by 7, the original number is too. (e.g., for 343 -> 34 - (2*3) = 28, which is divisible by 7).
By 11: If the difference between the sum of digits at odd places and the sum of digits at even places is 0 or a multiple of 11.

2. Important Theorems for Remainders:

Remainder Theorem: The remainder of \( \frac{f(x)}{x-a} \) is \( f(a) \).
Fermat's Little Theorem: If 'p' is a prime number, then \( a^{p-1} \) divided by 'p' gives a remainder of 1. (Here 'a' should not be a multiple of 'p').
Euler's Totient Theorem: A powerful generalization of Fermat's theorem used for composite divisors.

Combined Divisibility: To check for divisibility by a composite number (e.g., 72), check for divisibility by its co-prime factors (8 and 9).
Negative Remainders: Sometimes, using a negative remainder simplifies calculations. The remainder of 15 divided by 8 is 7, which can also be thought of as -1. This is very useful in power calculations. Example: \( 15^{100} \div 8 \) gives a remainder of \( (-1)^{100} \), which is 1.
Cyclicity of Remainders: The remainders of powers of a number often repeat in a cycle. Find the cycle length to solve questions with very large powers quickly. For example, the unit digits of powers of 2 are (2, 4, 8, 6, 2, ...), a cycle of length 4.

Memorize Thoroughly: The divisibility rules are non-negotiable. Write them down and practice until they are second nature.
Eliminate Options: Use divisibility rules as a fast first-pass filter to eliminate incorrect choices in MCQs.
Understand, Don't Just Memorize: Focus on how and why a theorem like Fermat's works. This allows you to apply it correctly even in tricky questions.
Practice with Variety: Solve a wide range of problems to get comfortable with different patterns and question types involving remainders.