Only positive integer factors are listed. The calculator does not enumerate negative pairs, factor algebraic expressions or polynomials, or compare a collection of numbers.
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Enter one positive integer to see its complete factor list, matching factor pairs, prime factorization, and every split in the factor tree.
Calculated 16 positive factors of 120.
The prime leaves multiply back to 120: 2 × 2 × 2 × 3 × 5.
Factor-pair workshop
Factors answer a divisibility question: which whole numbers multiply exactly to the target? Reading factor pairs as rectangular arrangements makes the list tangible and leads naturally to prime factorization, common factors, and multiples.
Step-by-step guide · Factor-pair workshop
Begin 84 = 7×12. Seven is prime, while 12 can be decomposed further. A different first split, such as 4×21, must eventually reach the same prime factors.
Write 12 = 3×4 and 4 = 2×2. The leaves are 7, 3, 2, and 2, so the compact prime factorization is 2²×3×7.
Calculate 4×3×7 = 84. This reconstruction check catches a missing repeated prime or an unfinished composite branch.
A coordinator has 84 identical cards and wants equal rows with none left over. Testing positive divisors yields the pairs 1×84, 2×42, 3×28, 4×21, 6×14, and 7×12. Each pair describes a valid rectangle. Ten rows fail because 84 ÷ 10 leaves a remainder, so 10 is not a factor even though it is numerically close to √84.
Once a divisor passes √84, its partner has already appeared in an earlier pair. That symmetry explains the efficient search boundary: if 84 had a factor larger than its square root without a smaller partner, their product would exceed 84. The complete positive factor list is 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84. Negative factor pairs may matter in algebra, but a physical grouping task normally asks for positive counts.
Begin 84 = 7×12. Seven is prime, while 12 can be decomposed further. A different first split, such as 4×21, must eventually reach the same prime factors.
Write 12 = 3×4 and 4 = 2×2. The leaves are 7, 3, 2, and 2, so the compact prime factorization is 2²×3×7.
Calculate 4×3×7 = 84. This reconstruction check catches a missing repeated prime or an unfinished composite branch.
Use this page to understand one target such as 84. A pair and its reversed order are one structural relationship, not two different factorizations.
If 84 cards and 126 envelopes must be split into the greatest number of identical kits, use the greatest common factor calculator. It intersects factor structure across quantities.
If one task repeats every 84 minutes and another every 126 minutes, the next shared time comes from the least common multiple calculator, not from the largest shared divisor.