Number Properties & Divisibility Analyzer
Analyze any integer instantaneously: Parity (even/odd), primality, prime factors, complete divisor list, power roots, digit reversal, palindrome detection, and base conversions.
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Interactive Modulo & Division Workbench
Core Principles of Integer Properties & Divisibility
1. Parity (Even vs. Odd Numbers)
An integer $n$ is even if it is divisible by 2 without a remainder ($n = 2k$ for some integer $k$). Conversely, an integer is odd if dividing by 2 leaves a remainder of 1 ($n = 2k + 1$). Zero is classified as an even integer because $0 = 2 \times 0$.
2. Prime Numbers & The Fundamental Theorem of Arithmetic
A prime number is an integer strictly greater than 1 that possesses exactly two distinct positive divisors: 1 and itself. The Fundamental Theorem of Arithmetic dictates that every integer greater than 1 can be uniquely represented as a product of prime powers, up to the order of the factors.
3. The Euclidean Division Lemma (Modulo Operation)
For any positive dividend $a$ and divisor $b$, there exist unique integers $q$ (quotient) and $r$ (remainder) such that:
a = b × q + r, where 0 ≤ r < b.
The modulo operator calculates $r = a \pmod b$.
Quick Divisibility Rules Cheat Sheet
| Divisor | Test Rule | Example |
|---|---|---|
| 2 | Last digit is even (0, 2, 4, 6, 8) | 1,354 ends in 4 |
| 3 | Sum of all digits is divisible by 3 | 456 → 4+5+6 = 15 |
| 4 | Last two digits form a number divisible by 4 | 1,324 → 24 ÷ 4 = 6 |
| 5 | Last digit is either 0 or 5 | 925 ends in 5 |
| 6 | Number is divisible by both 2 and 3 | 342 is even and 3+4+2=9 |
| 9 | Sum of all digits is divisible by 9 | 729 → 7+2+9 = 18 |
| 11 | Alternating sum of digits is a multiple of 11 | 1,331 → (1+3)-(3+1)=0 |
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