Algebra and number theory calculators solve some of the most common problems in arithmetic, combinatorics, and pre-algebra: breaking numbers into their prime building blocks, counting arrangements, simplifying fractions, and expressing extreme values compactly. These six tools cover the core toolkit that shows up in classroom homework, competitive exam prep, probability problems, and even everyday programming tasks like hashing and cryptography. Each one is built around a specific, well-defined formula rather than an approximation, so results are exact.
This guide walks through each calculator in the order you are most likely to need them — starting with factorial, which underlies permutations and combinations, moving through GCF/LCM and prime factorization, and finishing with root extraction and scientific notation for handling numbers at either extreme of scale. Along the way, each step includes a worked example so you can check your own hand calculation against the formula before reaching for the calculator.
Step 1: Calculate Factorials for Permutations and Sequences
The factorial of a non-negative integer n, written n!, is the product of every positive integer from n down to 1: n! = n × (n−1) × (n−2) × ... × 2 × 1. By definition, 0! = 1, since there is exactly one way to arrange zero items. Factorials grow extremely fast — 5! is 120, but 10! is already 3,628,800, and 15! exceeds one trillion.
Factorials show up any time you need to count the number of ways to arrange a full set of distinct items in order. If you have 6 books and want to know how many different ways you can line them up on a shelf, the answer is 6! = 720. This is also the foundation of permutations and combinations, both of which are built directly from factorial ratios (covered in Step 3).
Factorials also appear in series expansions used across calculus and physics, such as the Taylor series for eˣ, which sums terms of the form xⁿ/n!. In probability, factorials calculate the total sample space size before you apply constraints — for example, the number of possible orderings of a 52-card deck is 52!, a number with 68 digits.
Worked example: Calculate 7!. 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 42 × 5 × 4 × 3 × 2 × 1 = 210 × 4 × 3 × 2 × 1 = 840 × 3 × 2 × 1 = 2,520 × 2 × 1 = 5,040
So 7! = 5,040 — meaning there are 5,040 distinct ways to arrange 7 unique items in a row. The Factorial Calculator computes this instantly for any non-negative integer and shows the full step-by-step multiplication chain, which is useful for checking your arithmetic on larger values where a manual error is easy to make.
One practical caution: factorial values overflow standard number precision quickly. Past roughly 170!, the result exceeds what a standard 64-bit floating-point number can represent exactly, so if you are working with extremely large factorials in a spreadsheet or script, expect precision loss beyond that point.
Step 2: Find the GCF and LCM of Two Numbers
The greatest common factor (GCF), also called the greatest common divisor (GCD), is the largest positive integer that divides two numbers with no remainder. The least common multiple (LCM) is the smallest positive integer that both numbers divide into evenly. Both concepts are essential for working with fractions: GCF reduces fractions to lowest terms, and LCM finds a common denominator for addition and subtraction.
The most efficient way to compute GCF for large numbers is the Euclidean algorithm: repeatedly replace the larger number with the remainder of dividing it by the smaller number, until the remainder reaches zero — the last non-zero remainder is the GCF. For example, GCF(48, 18): 48 mod 18 = 12, then 18 mod 12 = 6, then 12 mod 6 = 0, so the GCF is 6. This method avoids the need to fully factor either number, which matters for large values.
An equally valid approach for smaller numbers is prime factorization: break both numbers into their prime factors, then take the lowest power of each shared prime for the GCF, and the highest power of every prime appearing in either number for the LCM.
Worked example: Find the GCF and LCM of 24 and 36. Prime factorization: 24 = 2³ × 3¹, and 36 = 2² × 3². GCF: take the lower power of each shared prime → 2² × 3¹ = 4 × 3 = 12. LCM: take the higher power of each prime present → 2³ × 3² = 8 × 9 = 72.
Check: GCF × LCM should equal the product of the two original numbers. 12 × 72 = 864, and 24 × 36 = 864 — they match. This identity, GCF(a,b) × LCM(a,b) = a × b, always holds and is a fast way to verify your answer or compute one value once you already know the other. The GCF and LCM Calculator returns both values instantly for any pair of integers, along with the prime factorization used to derive them.
Step 3: Compute Permutations and Combinations
Permutations and combinations both count the number of ways to select r items from a group of n items, but they differ in one crucial respect: permutations count order, combinations do not. The permutation formula is nPr = n! ÷ (n−r)!, and the combination formula is nCr = n! ÷ (r! × (n−r)!). Notice that nCr = nPr ÷ r! — combinations are simply permutations divided by the number of ways to reorder the r selected items, since each unique group counts once instead of r! times.
Deciding which formula applies comes down to a single question: does the order of selection change the outcome? Ranking the top 3 finishers in a race (1st, 2nd, 3rd) is a permutation problem, because swapping two finishers produces a different result. Selecting 3 people for a project team with no distinct roles is a combination problem, because the same 3 people form the same team regardless of the order they were chosen in.
Worked example: From a group of 8 people, how many ways can you choose a president, secretary, and treasurer (3 distinct roles), and how many ways can you choose an unordered 3-person committee?
Permutations (order matters — roles are distinct): 8P3 = 8! ÷ (8−3)! = 8! ÷ 5! = (8 × 7 × 6 × 5!) ÷ 5! = 8 × 7 × 6 = 336.
Combinations (order doesn't matter — committee has no roles): 8C3 = 8! ÷ (3! × 5!) = 336 ÷ 6 = 56.
The relationship holds: 336 ÷ 3! = 336 ÷ 6 = 56. The Permutation and Combination Calculator computes both nPr and nCr for any n and r in one entry, which is useful for confirming which formula applies to a specific word problem by comparing both outputs.
Step 4: Break a Number Down with Prime Factorization
Prime factorization expresses any integer greater than 1 as a product of prime numbers raised to whole-number powers. Every integer has exactly one prime factorization — this is the Fundamental Theorem of Arithmetic — which makes prime factorization a foundational building block for GCF, LCM, simplifying radicals, and testing primality.
The standard method is trial division: divide the number by the smallest prime (2) repeatedly until it no longer divides evenly, then move to the next prime (3, 5, 7, 11, ...), continuing until the remaining value is 1. You only need to test primes up to the square root of the original number — if nothing up to that point divides evenly, whatever remains is itself prime.
Worked example: Find the prime factorization of 360. 360 ÷ 2 = 180 180 ÷ 2 = 90 90 ÷ 2 = 45 (no longer divisible by 2) 45 ÷ 3 = 15 15 ÷ 3 = 5 (no longer divisible by 3) 5 is itself prime.
So 360 = 2³ × 3² × 5¹. From this factorization, you can immediately compute the total number of divisors by adding 1 to each exponent and multiplying: (3+1) × (2+1) × (1+1) = 4 × 3 × 2 = 24 divisors. The Prime Factorization Calculator handles this automatically for any integer, reporting the full factorization, the total divisor count, and whether the input itself is prime — useful for quickly checking large numbers by hand would otherwise take a while to verify.
Step 5: Calculate Square Roots, Cube Roots, and Nth Roots
The nth root of a number x is the value that, when raised to the power n, produces x. This is written as ⁿ√x, and it is mathematically equivalent to raising x to the power of 1/n. The square root (n = 2) and cube root (n = 3) are the most common cases, but any positive integer degree is valid, including fourth, fifth, and sixth roots.
Square roots and other even-degree roots behave differently from odd-degree roots when it comes to negative numbers. An odd root of a negative number is well-defined within the real numbers — the cube root of −8 is −2, because (−2)³ = −8. An even root of a negative number, however, has no real solution, because no real number squared (or raised to any even power) can produce a negative result. This is why root calculators that only handle real numbers typically restrict input to non-negative values for even-degree roots.
Worked example: Find the fourth root of 81. ⁸¹^(1/4) — look for a value y such that y⁴ = 81. Testing y = 3: 3⁴ = 3 × 3 × 3 × 3 = 81. So the fourth root of 81 is 3.
Worked example: Find the cube root of 125. Looking for y such that y³ = 125. Testing y = 5: 5³ = 125. So the cube root of 125 is 5.
The Root Calculator computes square roots, cube roots, and roots up to the sixth degree for any non-negative value, which is particularly useful for degrees beyond square and cube, where mental estimation becomes impractical (finding the fifth root of 100,000 by hand, for instance, is not something most people can do quickly).
Step 6: Convert Numbers to and from Scientific Notation
Scientific notation expresses a number as a coefficient between 1 and 10 (inclusive of 1, exclusive of 10) multiplied by a power of 10: a × 10ᵇ, where 1 ≤ |a| < 10. It exists to make extremely large or extremely small numbers easier to write, compare, and calculate with, without tracking long strings of zeros.
To convert a standard decimal into scientific notation, move the decimal point until exactly one non-zero digit remains before it, and count the number of places moved — that count becomes the exponent. Moving the decimal to the left (for large numbers) produces a positive exponent; moving it to the right (for small numbers, less than 1) produces a negative exponent.
Worked example: Convert 6,371,000 to scientific notation. Move the decimal point left until one digit remains before it: 6,371,000 → 6.371, moved 6 places. Result: 6.371 × 10⁶.
Worked example: Convert 0.000047 to scientific notation. Move the decimal point right until one non-zero digit remains before it: 0.000047 → 4.7, moved 5 places right, giving a negative exponent. Result: 4.7 × 10⁻⁵.
To convert back from scientific notation to a decimal, reverse the process: shift the decimal point by the number of places indicated by the exponent, in the direction matching its sign. The Scientific Notation Calculator converts in both directions and is especially useful for double-checking unit conversions in physics and chemistry, where values routinely span from atomic scales (10⁻¹⁹ and smaller) to astronomical scales (10²³ and larger).
Key Terms
- Factorial (n!) — the product of all positive integers from n down to 1; 0! is defined as 1.
- GCF (Greatest Common Factor) — the largest integer that divides two or more numbers without a remainder; also called GCD.
- LCM (Least Common Multiple) — the smallest positive integer that two or more numbers divide into evenly.
- Euclidean Algorithm — a method for finding the GCF of two numbers through repeated division and remainder-taking, without needing to fully factor either number.
- Permutation (nPr) — a count of ordered arrangements of r items chosen from n, where order matters.
- Combination (nCr) — a count of unordered selections of r items chosen from n, where order does not matter.
- Prime Factorization — expressing an integer as a product of prime numbers raised to whole-number powers.
- Nth Root — the value which, raised to the power n, produces a given number; equivalent to raising that number to the power 1/n.
- Scientific Notation — a way of writing numbers as a coefficient between 1 and 10 multiplied by a power of 10, used to compactly represent very large or very small values.