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Number system formulas
16 formulas for Number system, from the Arithmetic & Quantitative Aptitude section of the Quantix formula library. Tap 🧮 next to a formula to solve it for any variable with step-by-step working.
| Name | Formula | Note |
|---|---|---|
| Natural numbers | ℕ = {1, 2, 3, …} | |
| Whole numbers | W = {0, 1, 2, 3, …} | |
| Integers | ℤ = {…, −2, −1, 0, 1, 2, …} | |
| Rational number | p / q, q ≠ 0, p, q ∈ ℤ | terminating or repeating decimal |
| Sum of first n natural numbers | 1 + 2 + … + n = n(n + 1) / 2 | |
| Sum of squares | 1² + 2² + … + n² = n(n + 1)(2n + 1) / 6 | |
| Sum of cubes | 1³ + 2³ + … + n³ = [n(n + 1) / 2]² | |
| Sum of first n odd numbers | 1 + 3 + 5 + … + (2n − 1) = n² | |
| Sum of first n even numbers | 2 + 4 + … + 2n = n(n + 1) | |
| Number of factors | N = p^a · q^b · r^c ⇒ factors = (a + 1)(b + 1)(c + 1) | p, q, r primes |
| Sum of factors | (p^(a+1) − 1)/(p − 1) · (q^(b+1) − 1)/(q − 1) · … | |
| Number of primes below 100 | 25 | 2, 3, 5, 7, 11, …, 97 |
| Dividend relation | Dividend = Divisor × Quotient + Remainder | |
| Unit digit cycle | 2, 3, 7, 8 repeat every 4 powers; 4, 9 every 2; 0, 1, 5, 6 constant | use exponent mod 4 |
| Euler's totient | φ(n) = n(1 − 1/p₁)(1 − 1/p₂)… | p = distinct prime factors of n |
| Trailing zeros of n! | ⌊n/5⌋ + ⌊n/25⌋ + ⌊n/125⌋ + … |
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