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What is the total number of divisors of 1728?

Find the number of divisors of 1728


Method:
Find  the prime factorization of the number n.
Take all the exponents in the factorization.
Add 1 to each
Then multiply these "exponents + 1"s together.

Solution:

Prime Factorization of 1728:

2 | 1728 
2 | 864

2 | 432

2 | 216  

2 | 108

2 | 54

3 | 27

3 | 9

3 | 3

   | 1

2^6, 3^3
Exponents: 6 & 3

Add 1 to each exponent (6+1) (3+1)
Multiple “exponents+1” together = (6+1) x (3+1) =7 x 4 = 28

Total number of divisors of 1728 = 28

What is the total number of divisors of 600?

Numbers: Find the number of divisors of 600

Method:
Find  the prime factorization of the number n.
Take all the exponents in the factorization.
Add 1 to each
Then multiply these "exponents + 1"s together.

Example:
Find the number of divisors of 600

Solution:
Prime Factorization of 600:

2 | 600 
2 | 300
2 | 150 
3 | 75
5 | 25
5 | 5
   | 1 
 
2^3, 3^1 & 5^2

Exponents: 3, 1 & 2
Add 1 to each exponent = (3+1) (1+1) (2+1)
Multiple “exponents+1” together = (3+1) x (1+1) x (2+1) = 4 x 2 x 3 = 24

Total number of divisors of 600 = 24

Find the sum of divisors of 544

Find the sum of divisors of 544.

Method:
Find the the prime factorization of the number
Calculate the sum of its divisors
Take each different prime factor
Add together all its powers up to the one that appears in the prime factorization
And then multiply all these sums together!

Solution:

Prime Factorization of 544:

2  | 544  

2  | 272

2  | 136

2  | 68

2  | 34
17 | 17

     | 1

2^5  x   17^1
Exponents: 5 & 1

X	2^0	2^1	2^2	2^3	2^4	2^5	Row Sum
17^0	1	2	4	8	16	32	63
17^1	17	34	68	136	272	544	1071
Col  Sum	18	36	72	144	288	576	1134

Sum of divisors of 544 = 1134

Find the sum of divisors of 544 excluding 1 and 544

Find the sum of divisors of 544 excluding 1 and 544

Method:
Find the the prime factorization of the number
Calculate the sum of its divisors
Take each different prime factor
Add together all its powers up to the one that appears in the prime factorization
And then multiply all these sums together
Deduct the numbers to be excluded from the result

Solution:

Prime Factorization of 544:

2  | 544  

2  | 272

2  | 136

2  | 68

2  | 34
17 | 17

    | 1

2^5 x 17^1   
Exponents: 5 & 1

X	2^0	2^1	2^2	2^3	2^4	2^5	Row Sum
17^0	1	2	4	8	16	32	63
17^1	17	34	68	136	272	544	1071
Col  Sum	18	36	72	144	288	576	1134

Sum of divisors of 544 excluding 1 and 544 = 1134 -1 - 544 = 589

Answer: 589