Prime Factorization Calculator

Understanding Prime Numbers

A prime number is a natural number greater than 1 that possesses exactly two distinct divisors: 1 and itself. For instance, 5 satisfies this condition because no other whole number divides it exactly. However, 6 fails the test since 2 and 3 also evenly divide it, making it a composite number. The number 1 is deliberately excluded from the set of primes because it has only one divisor. This definition, solidified in the early 20th century, ensures uniqueness in factorization.

Prime Factors and Prime Factorization

Prime factors are the prime numbers that multiply together to yield the original number. Take 20: its factor pairs are (1,20), (2,10), and (4,5). Among these, only 2 and 5 are prime, so they are the prime factors.

Prime factorization is the process of rewriting a number solely as a product of prime factors. Following the same example: $20 = 2 \times 2 \times 5 = 2^{2} \times 5$. With the Prime Factorization Calculator, you can obtain this decomposition instantly for any integer.

Using the Factor Tree Method

A factor tree offers a visual approach to prime factorization. Let’s run through the steps with 36.

1. Start with 36 at the top.
2. Split it into two branches: for example, 6 and 6.
3. Factor each 6 into 2 and 3. Both are prime, so the branches end.
4. Collect all terminal leaves: 2, 2, 3, 3.
5. Multiply: $36 = 2 \times 2 \times 3 \times 3 = 2^{2} \times 3^{2}$.

The Factor Tree Calculator built into this tool draws the tree for you and can accept alternative splits (e.g., 4 × 9 or 2 × 18) while always producing the same final factorization.

Now consider 72, a slightly richer example:

• 72 splits into 8 × 9.
• 8 decomposes to $2^{3}$; 9 to $3^{2}$.
• Hence $72 = 2^{3} \times 3^{2}$.

Another commonly queried number is 100. The prime factorization of 100 is obtained by splitting 100 into 10 × 10, then factoring each 10: $100 = 2 \times 5 \times 2 \times 5 = 2^{2} \times 5^{2}$.

Applications: Greatest Common Factor and Least Common Multiple

Prime factorization provides a systematic way to compute the greatest common factor (GCF) and the least common multiple (LCM).

• GCF: For 6 and 20, write $6 = 2 \times 3$ and $20 = 2^{2} \times 5$. The only common prime is 2, so GCF = 2. This method simplifies fractions that involve these numbers.
• LCM: Multiply the highest power of each prime appearing in either factorization: $2^{2} \times 3 \times 5 = 60$. The LCM is crucial when adding fractions with unlike denominators.

Many online calculators, including this free prime factor finder online, derive the GCF and LCM from the prime factorization automatically.

Reference Table: Prime Factorizations of Integers 2–100

The following table lists the prime factorization for every number from 2 through 100. Each entry can be verified with the Prime Factorization Calculator.

Number	Prime Factorization
2	$2$ (prime)
3	$3$ (prime)
4	$2^{2}$
5	$5$ (prime)
6	$2 \times 3$
7	$7$ (prime)
8	$2^{3}$
9	$3^{2}$
10	$2 \times 5$
11	$11$ (prime)
12	$2^{2} \times 3$
13	$13$ (prime)
14	$2 \times 7$
15	$3 \times 5$
16	$2^{4}$
17	$17$ (prime)
18	$2 \times 3^{2}$
19	$19$ (prime)
20	$2^{2} \times 5$
21	$3 \times 7$
22	$2 \times 11$
23	$23$ (prime)
24	$2^{3} \times 3$
25	$5^{2}$
26	$2 \times 13$
27	$3^{3}$
28	$2^{2} \times 7$
29	$29$ (prime)
30	$2 \times 3 \times 5$
31	$31$ (prime)
32	$2^{5}$
33	$3 \times 11$
34	$2 \times 17$
35	$5 \times 7$
36	$2^{2} \times 3^{2}$
37	$37$ (prime)
38	$2 \times 19$
39	$3 \times 13$
40	$2^{3} \times 5$
41	$41$ (prime)
42	$2 \times 3 \times 7$
43	$43$ (prime)
44	$2^{2} \times 11$
45	$3^{2} \times 5$
46	$2 \times 23$
47	$47$ (prime)
48	$2^{4} \times 3$
49	$7^{2}$
50	$2 \times 5^{2}$
51	$3 \times 17$
52	$2^{2} \times 13$
53	$53$ (prime)
54	$2 \times 3^{3}$
55	$5 \times 11$
56	$2^{3} \times 7$
57	$3 \times 19$
58	$2 \times 29$
59	$59$ (prime)
60	$2^{2} \times 3 \times 5$
61	$61$ (prime)
62	$2 \times 31$
63	$3^{2} \times 7$
64	$2^{6}$
65	$5 \times 13$
66	$2 \times 3 \times 11$
67	$67$ (prime)
68	$2^{2} \times 17$
69	$3 \times 23$
70	$2 \times 5 \times 7$
71	$71$ (prime)
72	$2^{3} \times 3^{2}$
73	$73$ (prime)
74	$2 \times 37$
75	$3 \times 5^{2}$
76	$2^{2} \times 19$
77	$7 \times 11$
78	$2 \times 3 \times 13$
79	$79$ (prime)
80	$2^{4} \times 5$
81	$3^{4}$
82	$2 \times 41$
83	$83$ (prime)
84	$2^{2} \times 3 \times 7$
85	$5 \times 17$
86	$2 \times 43$
87	$3 \times 29$
88	$2^{3} \times 11$
89	$89$ (prime)
90	$2 \times 3^{2} \times 5$
91	$7 \times 13$
92	$2^{2} \times 23$
93	$3 \times 31$
94	$2 \times 47$
95	$5 \times 19$
96	$2^{5} \times 3$
97	$97$ (prime)
98	$2 \times 7^{2}$
99	$3^{2} \times 11$
100	$2^{2} \times 5^{2}$

Examples Beyond 100

The same logic applies to larger numbers. Below are factorizations of selected values between 101 and 250, plus a few beyond:

• 101 (prime), 102 = $2 \times 3 \times 17$, 104 = $2^{3} \times 13$, 105 = $3 \times 5 \times 7$, 108 = $2^{2} \times 3^{3}$, 117 = $3^{2} \times 13$, 120 = $2^{3} \times 3 \times 5$, 121 = $11^{2}$, 125 = $5^{3}$, 126 = $2 \times 3^{2} \times 7$, 130 = $2 \times 5 \times 13$, 132 = $2^{2} \times 3 \times 11$, 135 = $3^{3} \times 5$, 140 = $2^{2} \times 5 \times 7$, 144 = $2^{4} \times 3^{2}$, 147 = $3 \times 7^{2}$, 150 = $2 \times 3 \times 5^{2}$, 162 = $2 \times 3^{4}$, 175 = $5^{2} \times 7$, 180 = $2^{2} \times 3^{2} \times 5$, 196 = $2^{2} \times 7^{2}$, 200 = $2^{3} \times 5^{2}$, 216 = $2^{3} \times 3^{3}$, 225 = $3^{2} \times 5^{2}$, 240 = $2^{4} \times 3 \times 5$, 243 = $3^{5}$, 250 = $2 \times 5^{3}$, 256 = $2^{8}$, 300 = $2^{2} \times 3 \times 5^{2}$, 360 = $2^{3} \times 3^{2} \times 5$, 400 = $2^{4} \times 5^{2}$, 441 = $3^{2} \times 7^{2}$, 500 = $2^{2} \times 5^{3}$, 512 = $2^{9}$, 525 = $3 \times 5^{2} \times 7$, 540 = $2^{2} \times 3^{3} \times 5$, 576 = $2^{6} \times 3^{2}$, 600 = $2^{3} \times 3 \times 5^{2}$, 625 = $5^{4}$.

These examples illustrate how numbers break down into primes; the Prime Factor Calculator can handle any integer, yielding a complete factorization in seconds.

A Note on the Number 1

Historically, 1 was classified as a prime number until the early twentieth century, when mathematicians redefined primes to require exactly two distinct divisors. Consequently, 1 is not included among the prime factors output by the calculator.

How the Calculator Works

Enter a number into the Prime Factorization Calculator (also a Free Prime Factor Finder Online), and the tool performs trial division by successive primes, stopping when the quotient is 1. It displays the factorization both as a product of primes and in exponential form, with an optional factor tree diagram. This makes the calculator equally useful as a Factor Tree Calculator for educational purposes.

Summary

Prime factorization is a fundamental concept that underpins many areas of mathematics, from fraction simplification to encryption. The free Prime Factorization tool on this page delivers accurate decompositions instantly, supporting learning and problem‑solving. Whether you need the prime factorization of 100, want to explore factor trees, or simply require a reliable Prime Factor Calculator, this resource is designed to meet your needs.