Calculate and Count All the Factors of 448,448. Online Calculator

All the factors (divisors) of the number 448,448. How important is the prime factorization of the number

1. Carry out the prime factorization of the number 448,448:

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


448,448 = 26 × 72 × 11 × 13
448,448 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.


How to count the number of factors of a number?

If a number N is prime factorized as:
N = am × bk × cz
where a, b, c are the prime factors and m, k, z are their exponents, natural numbers, ....


Then the number of factors of the number N can be calculated as:
n = (m + 1) × (k + 1) × (z + 1)


In our case, the number of factors is calculated as:

n = (6 + 1) × (2 + 1) × (1 + 1) × (1 + 1) = 7 × 3 × 2 × 2 = 84

But to actually calculate the factors, see below...

2. Multiply the prime factors of the number 448,448

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 = 2
22 = 4
prime factor = 7
23 = 8
prime factor = 11
prime factor = 13
2 × 7 = 14
24 = 16
2 × 11 = 22
2 × 13 = 26
22 × 7 = 28
25 = 32
22 × 11 = 44
72 = 49
22 × 13 = 52
23 × 7 = 56
26 = 64
7 × 11 = 77
23 × 11 = 88
7 × 13 = 91
2 × 72 = 98
23 × 13 = 104
24 × 7 = 112
11 × 13 = 143
2 × 7 × 11 = 154
24 × 11 = 176
2 × 7 × 13 = 182
22 × 72 = 196
24 × 13 = 208
25 × 7 = 224
2 × 11 × 13 = 286
22 × 7 × 11 = 308
25 × 11 = 352
22 × 7 × 13 = 364
23 × 72 = 392
25 × 13 = 416
26 × 7 = 448
72 × 11 = 539
22 × 11 × 13 = 572
23 × 7 × 11 = 616
72 × 13 = 637
This list continues below...

... This list continues from above
26 × 11 = 704
23 × 7 × 13 = 728
24 × 72 = 784
26 × 13 = 832
7 × 11 × 13 = 1,001
2 × 72 × 11 = 1,078
23 × 11 × 13 = 1,144
24 × 7 × 11 = 1,232
2 × 72 × 13 = 1,274
24 × 7 × 13 = 1,456
25 × 72 = 1,568
2 × 7 × 11 × 13 = 2,002
22 × 72 × 11 = 2,156
24 × 11 × 13 = 2,288
25 × 7 × 11 = 2,464
22 × 72 × 13 = 2,548
25 × 7 × 13 = 2,912
26 × 72 = 3,136
22 × 7 × 11 × 13 = 4,004
23 × 72 × 11 = 4,312
25 × 11 × 13 = 4,576
26 × 7 × 11 = 4,928
23 × 72 × 13 = 5,096
26 × 7 × 13 = 5,824
72 × 11 × 13 = 7,007
23 × 7 × 11 × 13 = 8,008
24 × 72 × 11 = 8,624
26 × 11 × 13 = 9,152
24 × 72 × 13 = 10,192
2 × 72 × 11 × 13 = 14,014
24 × 7 × 11 × 13 = 16,016
25 × 72 × 11 = 17,248
25 × 72 × 13 = 20,384
22 × 72 × 11 × 13 = 28,028
25 × 7 × 11 × 13 = 32,032
26 × 72 × 11 = 34,496
26 × 72 × 13 = 40,768
23 × 72 × 11 × 13 = 56,056
26 × 7 × 11 × 13 = 64,064
24 × 72 × 11 × 13 = 112,112
25 × 72 × 11 × 13 = 224,224
26 × 72 × 11 × 13 = 448,448

The final answer:
(scroll down)

448,448 has 84 factors (divisors):
1; 2; 4; 7; 8; 11; 13; 14; 16; 22; 26; 28; 32; 44; 49; 52; 56; 64; 77; 88; 91; 98; 104; 112; 143; 154; 176; 182; 196; 208; 224; 286; 308; 352; 364; 392; 416; 448; 539; 572; 616; 637; 704; 728; 784; 832; 1,001; 1,078; 1,144; 1,232; 1,274; 1,456; 1,568; 2,002; 2,156; 2,288; 2,464; 2,548; 2,912; 3,136; 4,004; 4,312; 4,576; 4,928; 5,096; 5,824; 7,007; 8,008; 8,624; 9,152; 10,192; 14,014; 16,016; 17,248; 20,384; 28,028; 32,032; 34,496; 40,768; 56,056; 64,064; 112,112; 224,224 and 448,448
out of which 4 prime factors: 2; 7; 11 and 13
448,448 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".