Given the Number 729,729 Calculate (Find) All Its Factors (Divisors – the Proper, the Improper and the Prime Factors). Online Calculator

All the factors (divisors) of the number 729,729

1. Carry out the prime factorization of the number 729,729:

The prime factorization of a number: finding the prime numbers that multiply together to make that number.


729,729 = 36 × 7 × 11 × 13
729,729 is not a prime number but a composite one.


* Prime number: a natural number that is divisible (divided evenly) only by 1 and itself. A prime number has exactly two factors: 1 and the number itself.
* Composite number: a natural number that has at least one other factor than 1 and itself.


2. Multiply the prime factors of the number 729,729

Multiply the prime factors involved in the prime factorization of the number in all their unique combinations, that give different results.


Also consider the exponents of these prime factors.

Also add 1 to the list of factors (divisors). All the numbers are divisible by 1.


All the factors (divisors) are listed below - in ascending order

The list of factors (divisors):

neither prime nor composite = 1
prime factor = 3
prime factor = 7
32 = 9
prime factor = 11
prime factor = 13
3 × 7 = 21
33 = 27
3 × 11 = 33
3 × 13 = 39
32 × 7 = 63
7 × 11 = 77
34 = 81
7 × 13 = 91
32 × 11 = 99
32 × 13 = 117
11 × 13 = 143
33 × 7 = 189
3 × 7 × 11 = 231
35 = 243
3 × 7 × 13 = 273
33 × 11 = 297
33 × 13 = 351
3 × 11 × 13 = 429
34 × 7 = 567
32 × 7 × 11 = 693
36 = 729
32 × 7 × 13 = 819
This list continues below...

... This list continues from above
34 × 11 = 891
7 × 11 × 13 = 1,001
34 × 13 = 1,053
32 × 11 × 13 = 1,287
35 × 7 = 1,701
33 × 7 × 11 = 2,079
33 × 7 × 13 = 2,457
35 × 11 = 2,673
3 × 7 × 11 × 13 = 3,003
35 × 13 = 3,159
33 × 11 × 13 = 3,861
36 × 7 = 5,103
34 × 7 × 11 = 6,237
34 × 7 × 13 = 7,371
36 × 11 = 8,019
32 × 7 × 11 × 13 = 9,009
36 × 13 = 9,477
34 × 11 × 13 = 11,583
35 × 7 × 11 = 18,711
35 × 7 × 13 = 22,113
33 × 7 × 11 × 13 = 27,027
35 × 11 × 13 = 34,749
36 × 7 × 11 = 56,133
36 × 7 × 13 = 66,339
34 × 7 × 11 × 13 = 81,081
36 × 11 × 13 = 104,247
35 × 7 × 11 × 13 = 243,243
36 × 7 × 11 × 13 = 729,729

The final answer:
(scroll down)

729,729 has 56 factors (divisors):
1; 3; 7; 9; 11; 13; 21; 27; 33; 39; 63; 77; 81; 91; 99; 117; 143; 189; 231; 243; 273; 297; 351; 429; 567; 693; 729; 819; 891; 1,001; 1,053; 1,287; 1,701; 2,079; 2,457; 2,673; 3,003; 3,159; 3,861; 5,103; 6,237; 7,371; 8,019; 9,009; 9,477; 11,583; 18,711; 22,113; 27,027; 34,749; 56,133; 66,339; 81,081; 104,247; 243,243 and 729,729
out of which 4 prime factors: 3; 7; 11 and 13
729,729 and 1 are sometimes called improper factors, the others are called proper factors (proper divisors).

A quick way to find the factors (the divisors) of a number is to break it down into prime factors.


Then multiply the prime factors and their exponents, if any, in all their different combinations.


Factors (divisors), common factors (common divisors), the greatest common factor, GCF (also called the greatest common divisor, GCD, or the highest common factor, HCF)

  • If the number "t" is a factor (divisor) of the number "a" then in the prime factorization of "t" we will only encounter prime factors that also occur in the prime factorization of "a".
  • If there are exponents involved, the maximum value of an exponent for any base of a power that is found in the prime factorization of "t" (powers, or multiplicities) is at most equal to the exponent of the same base that is involved in the prime factorization of "a".
  • Hint: 23 = 2 × 2 × 2 = 8. 2 is called the base and 3 is the exponent. 23 is the power and 8 is the value of the power. We sometimes say that the number 2 is raised to the power of 3.
  • For example, 12 is a factor (divisor) of 120 - the remainder is zero when dividing 120 by 12.
  • Let's look at the prime factorization of both numbers and notice the bases and the exponents that occur in the prime factorization of both numbers:
  • 12 = 2 × 2 × 3 = 22 × 3
  • 120 = 2 × 2 × 2 × 3 × 5 = 23 × 3 × 5
  • 120 contains all the prime factors of 12, and all its bases' exponents are higher than those of 12.
  • If "t" is a common factor (divisor) of "a" and "b", then the prime factorization of "t" contains only the common prime factors involved in the prime factorizations of both "a" and "b".
  • If there are exponents involved, the maximum value of an exponent for any base of a power that is found in the prime factorization of "t" is at most equal to the minimum of the exponents of the same base that is involved in the prime factorization of both "a" and "b".
  • For example, 12 is the common factor of 48 and 360.
  • The remainder is zero when dividing either 48 or 360 by 12.
  • Here there are the prime factorizations of the three numbers, 12, 48 and 360:
  • 12 = 22 × 3
  • 48 = 24 × 3
  • 360 = 23 × 32 × 5
  • Please note that 48 and 360 have more factors (divisors): 2, 3, 4, 6, 8, 12, 24. Among them, 24 is the greatest common factor, GCF (or the greatest common divisor, GCD, or the highest common factor, HCF) of 48 and 360.
  • The greatest common factor, GCF, of two numbers, "a" and "b", is the product of all the common prime factors involved in the prime factorizations of both "a" and "b", taken by the lowest exponents.
  • Based on this rule it is calculated the greatest common factor, GCF, (or the greatest common divisor GCD, HCF) of several numbers, as shown in the example below...
  • GCF, GCD (1,260; 3,024; 5,544) = ?
  • 1,260 = 22 × 32
  • 3,024 = 24 × 32 × 7
  • 5,544 = 23 × 32 × 7 × 11
  • The common prime factors are:
  • 2 - its lowest exponent (multiplicity) is: min.(2; 3; 4) = 2
  • 3 - its lowest exponent (multiplicity) is: min.(2; 2; 2) = 2
  • GCF, GCD (1,260; 3,024; 5,544) = 22 × 32 = 252
  • Coprime numbers:
  • If two numbers "a" and "b" have no other common factors (divisors) than 1, gfc, gcd, hcf (a; b) = 1, then the numbers "a" and "b" are called coprime (or relatively prime).
  • Factors of the GCF
  • If "a" and "b" are not coprime, then every common factor (divisor) of "a" and "b" is a also a factor (divisor) of the greatest common factor, GCF (greatest common divisor, GCD, highest common factor, HCF) of "a" and "b".