Problem Solving

ardz24 wrote:The positive integer 200 has how many factors?

A. 2
B. 10
C. 12
D. 15
E. 24

I'm confused how to set up the formulas here. Can any experts help?

APPROACH #1 - FORMULA
If the prime factorization of N = (p^a)(q^b)(r^c) . . . (where p, q, r, etc are different prime numbers), then N has a total of (a+1)(b+1)(c+1)(etc) positive divisors.

Example: 14000 = (2^4)(5^3)(7^1)
So, the number of positive divisors of 14000 = (4+1)(3+1)(1+1) =(5)(4)(2) = 40

Now onto the question...
Example: 200 = (2^3)(5^2)
So, the number of positive divisors of 200 = (3+1)(2+1)
= (4)(3)
= 12
= C

APPROACH #2 - LIST
We can quickly list all of the factors of 200
I suggest we do so in PAIRS of values whose product is 200
We get:
1 and 200
2 and 100
4 and 50
5 and 40
8 and 25
10 and 20
DONE!

Total = 12
= C

Cheers,
Brent

BTGmoderatorAT wrote:The positive integer 200 has how many factors?

A. 2
B. 10
C. 12
D. 15
E. 24

I'm confused how to set up the formulas here. Can any experts help?

We can break 200 into primes, then add 1 to each exponent and find the product of all the sums. That product will give us the total number of factors.

200 = 20 x 10 = 2^2 x 5^1 x 2^1 x 5^1 = 2^3 x 5^2

Thus, 200 has (3 + 1)(2 + 1) = 4 x 3 = 12 factors.

(Note: It is not clear whether the question expects us to include negative factors as well. If negative factors are also included, the total number of factors will be twice the number of positive factors, which is 24.)

Answer: C