145,824: All the proper, improper and prime factors (divisors) of number

The fastest way to find all the factors (divisors) of 145,824: 1) Build its prime factorization & 2) Try out all the combinations of the prime factors that give different results

Note:

Factor of a number A: a number B that when multiplied with another C produces the given number A. Both B and C are factors of A.



Integer prime factorization:

Prime Factorization of a number: finding the prime numbers that multiply together to make that number.


145,824 = 25 × 3 × 72 × 31;
145,824 is not a prime, is a composite number;


* Positive integers that are only dividing by themselves and 1 are called prime numbers. A prime number has only two factors: 1 and itself.
* A composite number is a positive integer that has at least one factor (divisor) other than 1 and itself.




How to find all the factors (divisors) of the number?

145,824 = 25 × 3 × 72 × 31


Get all the combinations (multiplications) of the prime factors of the number that give different results.


When combining the prime factors also consider their exponents.


Also add 1 to the list of factors (divisors). Any number is divisible by 1.


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



Factors (divisors) list:

neither a prime nor a composite = 1
prime factor = 2
prime factor = 3
22 = 4
2 × 3 = 6
prime factor = 7
23 = 8
22 × 3 = 12
2 × 7 = 14
24 = 16
3 × 7 = 21
23 × 3 = 24
22 × 7 = 28
prime factor = 31
25 = 32
2 × 3 × 7 = 42
24 × 3 = 48
72 = 49
continued below...
... continued from above
23 × 7 = 56
2 × 31 = 62
22 × 3 × 7 = 84
3 × 31 = 93
25 × 3 = 96
2 × 72 = 98
24 × 7 = 112
22 × 31 = 124
3 × 72 = 147
23 × 3 × 7 = 168
2 × 3 × 31 = 186
22 × 72 = 196
7 × 31 = 217
25 × 7 = 224
23 × 31 = 248
2 × 3 × 72 = 294
24 × 3 × 7 = 336
22 × 3 × 31 = 372
23 × 72 = 392
2 × 7 × 31 = 434
24 × 31 = 496
22 × 3 × 72 = 588
3 × 7 × 31 = 651
25 × 3 × 7 = 672
23 × 3 × 31 = 744
24 × 72 = 784
22 × 7 × 31 = 868
25 × 31 = 992
23 × 3 × 72 = 1,176
2 × 3 × 7 × 31 = 1,302
24 × 3 × 31 = 1,488
72 × 31 = 1,519
25 × 72 = 1,568
23 × 7 × 31 = 1,736
24 × 3 × 72 = 2,352
22 × 3 × 7 × 31 = 2,604
25 × 3 × 31 = 2,976
2 × 72 × 31 = 3,038
24 × 7 × 31 = 3,472
3 × 72 × 31 = 4,557
25 × 3 × 72 = 4,704
23 × 3 × 7 × 31 = 5,208
22 × 72 × 31 = 6,076
25 × 7 × 31 = 6,944
2 × 3 × 72 × 31 = 9,114
24 × 3 × 7 × 31 = 10,416
23 × 72 × 31 = 12,152
22 × 3 × 72 × 31 = 18,228
25 × 3 × 7 × 31 = 20,832
24 × 72 × 31 = 24,304
23 × 3 × 72 × 31 = 36,456
25 × 72 × 31 = 48,608
24 × 3 × 72 × 31 = 72,912
25 × 3 × 72 × 31 = 145,824

Final answer:

145,824 has 72 factors:
1; 2; 3; 4; 6; 7; 8; 12; 14; 16; 21; 24; 28; 31; 32; 42; 48; 49; 56; 62; 84; 93; 96; 98; 112; 124; 147; 168; 186; 196; 217; 224; 248; 294; 336; 372; 392; 434; 496; 588; 651; 672; 744; 784; 868; 992; 1,176; 1,302; 1,488; 1,519; 1,568; 1,736; 2,352; 2,604; 2,976; 3,038; 3,472; 4,557; 4,704; 5,208; 6,076; 6,944; 9,114; 10,416; 12,152; 18,228; 20,832; 24,304; 36,456; 48,608; 72,912 and 145,824
out of which 4 prime factors: 2; 3; 7 and 31
145,824 (some consider that 1 too) is an improper factor (divisor), the others are proper factors (divisors).

The key to find the divisors of a number is to build its prime factorization.


Then determine all the different combinations (multiplications) of the prime factors, and their exponents, if any.



More operations of this kind:


Calculator: all the (common) factors (divisors) of numbers

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Tutoring: 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 "t" is a factor (divisor) of "a" then among the prime factors of "t" will appear only prime factors that also appear on the prime factorization of "a" and the maximum of their exponents (powers, or multiplicities) is at most equal to those involved in the prime factorization of "a".

For example, 12 is a factor (divisor) of 60:

  • 12 = 2 × 2 × 3 = 22 × 3
  • 60 = 2 × 2 × 3 × 5 = 22 × 3 × 5

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 both the prime factorizations of "a" and "b", by lower or at most by equal powers (exponents, or multiplicities).

For example, 12 is the common factor of 48 and 360. After running both numbers' prime factorizations (factoring them down to prime factors):

  • 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, is the product of all prime factors involved in both the prime factorizations of "a" and "b", by the lowest powers (multiplicities).

Based on this rule it is calculated the greatest common factor, GCF, (or greatest common divisor GCD, HCF) of several numbers, as shown in the example below:

  • 1,260 = 22 × 32;
  • 3,024 = 24 × 32 × 7;
  • 5,544 = 23 × 32 × 7 × 11;
  • Common prime factors are: 2 - its lowest power (multiplicity) is min.(2; 3; 4) = 2; 3 - its lowest power (multiplicity) is min.(2; 2; 2) = 2;
  • GCF, GCD (1,260; 3,024; 5,544) = 22 × 32 = 252;

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).

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".


What is a prime number?

What is a composite number?

Prime numbers up to 1,000

Prime numbers up to 10,000

Sieve of Eratosthenes

Euclid's algorithm

Simplifying ordinary (common) math fractions (reducing to lower terms): steps to follow and examples