Enter the first and second factors into the calculator to determine their product.
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Factor Pair Counts for Numbers 1 to 100
The number of factor pairs a number has reveals its divisor structure. Prime numbers always have exactly 1 factor pair (1 x p), while highly composite numbers have unusually many. The table below shows the positive factor pair count for key integers up to 100.
| Number | Factor Pairs | Count | Type |
|---|---|---|---|
| 1 | 1 x 1 | 1 | Unit |
| 2 | 1 x 2 | 1 | Prime |
| 6 | 1 x 6, 2 x 3 | 2 | Highly composite |
| 12 | 1 x 12, 2 x 6, 3 x 4 | 3 | Highly composite |
| 24 | 1 x 24, 2 x 12, 3 x 8, 4 x 6 | 4 | Highly composite |
| 36 | 1 x 36, 2 x 18, 3 x 12, 4 x 9, 6 x 6 | 5 | Perfect square, highly composite |
| 48 | 1 x 48, 2 x 24, 3 x 16, 4 x 12, 6 x 8 | 5 | Highly composite |
| 60 | 1 x 60, 2 x 30, 3 x 20, 4 x 15, 5 x 12, 6 x 10 | 6 | Highly composite |
| 72 | 1 x 72, 2 x 36, 3 x 24, 4 x 18, 6 x 12, 8 x 9 | 6 | Composite |
| 96 | 1 x 96, 2 x 48, 3 x 32, 4 x 24, 6 x 16, 8 x 12 | 6 | Composite |
| 100 | 1 x 100, 2 x 50, 4 x 25, 5 x 20, 10 x 10 | 5 | Perfect square |
Key pattern: the number 60 holds the record for most factor pairs under 100 (6 pairs), which is why ancient Babylonians chose base-60 for their number system. This same property makes 60 the basis for seconds in a minute and minutes in an hour.
Perfect Squares and Their Unique Factor Structure
Perfect squares always have an odd number of total divisors because one factor pair collapses into a single repeated factor (e.g., 6 x 6 = 36). Every other number has an even divisor count. This means that when you ask "what times what equals 36," you will find one pair where both factors are identical, while for a non-square like 35 (5 x 7), no such pair exists.
The perfect squares up to 400 and their collapsed factor pairs are: 1 (1x1), 4 (2x2), 9 (3x3), 16 (4x4), 25 (5x5), 36 (6x6), 49 (7x7), 64 (8x8), 81 (9x9), 100 (10x10), 121 (11x11), 144 (12x12), 169 (13x13), 196 (14x14), 225 (15x15), 256 (16x16), 289 (17x17), 324 (18x18), 361 (19x19), and 400 (20x20). The gap between consecutive perfect squares grows linearly: 4-1=3, 9-4=5, 16-9=7, 25-16=9, following the pattern of successive odd numbers.
Properties of Multiplication That Affect Factor Finding
Several algebraic properties govern how factor pairs behave and why the calculator handles certain inputs in specific ways.
Commutative property: a x b = b x a. This means 3 x 8 and 8 x 3 both equal 24. The calculator counts these as one pair rather than two, since order does not change the product.
Zero property: a x 0 = 0 for all values of a. This is why the factor pairs tab reports infinitely many pairs for the number 0. It also means the Solve 2 of 3 tab cannot determine a missing factor when the known factor is 0 and the product is also 0, since any number satisfies the equation.
Identity property: a x 1 = a. Every positive integer n has the trivial factor pair (1, n). A prime number is defined as a number greater than 1 whose only factor pair is this trivial one.
Sign rules: positive x positive = positive, negative x negative = positive, and positive x negative = negative. The calculator's Include negative pairs checkbox uses these rules to generate sign-flipped pairs for any target number.
The Divisor Function: Counting Factor Pairs at Scale
In number theory, the divisor function d(n) counts the total number of positive divisors of n. The number of distinct positive factor pairs equals d(n)/2 when n is not a perfect square, and (d(n)+1)/2 when it is. The first 20 values of d(n) are:
| n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| d(n) | 1 | 2 | 2 | 3 | 2 | 4 | 2 | 4 | 3 | 4 | 2 | 6 | 2 | 4 | 4 | 5 | 2 | 6 | 2 | 6 |
| Pairs | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 2 | 2 | 1 | 3 | 1 | 2 | 2 | 3 | 1 | 3 | 1 | 3 |
Notice that d(n) = 2 exactly when n is prime (divisors are only 1 and itself). The largest d(n) in this range is 6, which belongs to 12, 18, and 20. For very large numbers, the divisor function grows slowly: the average value of d(n) for numbers near N is approximately ln(N).
Common What Times What Equals Quick Reference
Below are the complete positive integer factor pairs for the most commonly searched target numbers. Use the calculator above for any number not listed here.
| Target | All Positive Factor Pairs | Prime? |
|---|---|---|
| 5 | 1 x 5 | Yes |
| 8 | 1 x 8, 2 x 4 | No |
| 10 | 1 x 10, 2 x 5 | No |
| 12 | 1 x 12, 2 x 6, 3 x 4 | No |
| 16 | 1 x 16, 2 x 8, 4 x 4 | No |
| 18 | 1 x 18, 2 x 9, 3 x 6 | No |
| 20 | 1 x 20, 2 x 10, 4 x 5 | No |
| 24 | 1 x 24, 2 x 12, 3 x 8, 4 x 6 | No |
| 25 | 1 x 25, 5 x 5 | No |
| 30 | 1 x 30, 2 x 15, 3 x 10, 5 x 6 | No |
| 36 | 1 x 36, 2 x 18, 3 x 12, 4 x 9, 6 x 6 | No |
| 42 | 1 x 42, 2 x 21, 3 x 14, 6 x 7 | No |
| 48 | 1 x 48, 2 x 24, 3 x 16, 4 x 12, 6 x 8 | No |
| 50 | 1 x 50, 2 x 25, 5 x 10 | No |
| 60 | 1 x 60, 2 x 30, 3 x 20, 4 x 15, 5 x 12, 6 x 10 | No |
| 72 | 1 x 72, 2 x 36, 3 x 24, 4 x 18, 6 x 12, 8 x 9 | No |
| 100 | 1 x 100, 2 x 50, 4 x 25, 5 x 20, 10 x 10 | No |
| 144 | 1 x 144, 2 x 72, 3 x 48, 4 x 36, 6 x 24, 8 x 18, 9 x 16, 12 x 12 | No |
Primes, Composites, and Why Factoring Matters
A prime number has exactly one positive factor pair: 1 times itself. Every other integer greater than 1 is composite, meaning it can be broken into at least two distinct factor pairs. The fundamental theorem of arithmetic guarantees that every integer greater than 1 has a unique prime factorization, and from that factorization, you can calculate the exact number of factor pairs without testing every possible divisor.
For a number with prime factorization p1^a1 x p2^a2 x ... x pk^ak, the total divisor count is (a1+1)(a2+1)...(ak+1). For example, 360 = 2^3 x 3^2 x 5^1, so d(360) = 4 x 3 x 2 = 24 divisors, which yields 12 factor pairs. This formula explains why numbers with many small prime factors (like 2, 3, and 5) tend to have the most factor pairs relative to their size.
The 25 primes under 100 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97. For any of these, asking what times what equals that prime has only one whole-number answer: 1 times the prime itself.
Where Factor Pairs Appear Outside Mathematics
Factor pairs show up whenever something needs to be divided evenly into rows and columns or groups of equal size. Arranging 24 chairs in a room gives 4 layout options: 1 row of 24, 2 rows of 12, 3 rows of 8, or 4 rows of 6. Packaging 60 items into boxes of uniform count yields 6 different box sizes. Screen resolution choices (1920 x 1080, 1280 x 720) are factor pairs of the total pixel count. Musical time signatures rely on how beats subdivide: 4/4 time divides a measure into 4 groups of 1, 2 groups of 2, or 1 group of 4.
In cryptography, the security of RSA encryption depends on the difficulty of finding the two prime factors of a very large composite number. A 2048-bit RSA key is the product of two primes each roughly 300 digits long. While the calculator above handles numbers up to 10 billion, factoring a 600-digit number with current technology would take longer than the age of the universe.
FAQ
What does what times what equals mean?
It asks for two numbers (factors) whose product is a given target. For example, what times what equals 36 asks for all pairs like 4 x 9, 6 x 6, 2 x 18, and so on. The calculator's Factor Pairs tab lists every positive integer pair, and optionally negative pairs as well.
How does the Solve 2 of 3 tab work?
Enter any two of the three values in a multiplication equation (factor x factor = product) and leave the third blank. The calculator solves for the missing value using division when needed. If both known values are 0, the missing value is indeterminate because any number times 0 equals 0.
Why do perfect squares have an odd number of divisors?
Divisors normally come in pairs (d, n/d). For perfect squares, the square root pairs with itself, adding one unpaired divisor to the total. For example, 36 has divisors 1, 2, 3, 4, 6, 9, 12, 18, 36 (nine divisors), where 6 is the unpaired middle divisor because 6 x 6 = 36.
Can this calculator handle decimals and negative numbers?
The Multiply and Solve 2 of 3 tabs fully support decimals and negative numbers. The Factor Pairs tab requires a whole number input because factor pairs are defined for integers, though negative pairs can be generated by checking the Include negative pairs box.
What is the largest number the calculator supports?
The Factor Pairs tab handles integers with an absolute value up to 10 billion (10,000,000,000). The Multiply and Solve tabs handle any number within JavaScript's floating-point precision range, roughly 15 to 17 significant digits.