1. What Is Divisibility?

In mathematics, divisibility refers to the ability of one integer to be divided by another integer without leaving any remainder. We say that integer "a" is divisible by integer "b" (b ≠ 0) if there exists an integer "k" such that a = b × k.

2. How The Divisibility Check Works

The calculator uses the modulo operation:

Where:

• \( a \) — The number being checked (dividend)
• \( b \) — The divisor (must be non-zero)
• \( \mod \) — The modulo operation that returns the remainder

Explanation: If dividing a by b leaves no remainder (remainder = 0), then a is divisible by b.

3. Importance Of Divisibility

Details: Divisibility rules are fundamental in number theory, cryptography, computer science algorithms, and many areas of mathematics. They help simplify fractions, factor numbers, and solve various mathematical problems.

4. Using The Calculator

Tips: Enter any integer for a (the number to check) and a positive integer for b (the divisor). The calculator will determine if a is divisible by b.

5. Frequently Asked Questions (FAQ)

Q1: Can zero be divisible by any number?
A: Yes, 0 is divisible by any non-zero integer because 0 ÷ b = 0 for any b ≠ 0.

Q2: Can a number be divisible by zero?
A: No, division by zero is undefined in mathematics.

Q3: What's the difference between a mod b and a ÷ b?
A: a ÷ b gives the quotient, while a mod b gives the remainder.

Q4: Are there quick divisibility rules for common numbers?
A: Yes, for example: divisible by 2 if last digit is even, by 3 if sum of digits is divisible by 3, by 5 if ends with 0 or 5.

Q5: How is divisibility used in real life?
A: Applications include checking digit validity (credit cards, barcodes), cryptography, scheduling repeating events, and distributing items equally.