sreak1089 wrote:45) If k is a positive integer, then 20k is divisible by how many different positive integers?
(1) k is prime
(2) k is 7
BACKGROUND INFO:
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....
Target question: 20k is divisible by how many different positive integers?
Statement 1: k is prime
There are several values of k that satisfy statement 1. Here are two:
Case a: k = 2, in which case 20k = (20)(2) = 40 = (2^
3)(5^
1). So, the number of divisors = (
3+1)(
1+1) = (4)(2) = 8. In other words,
20k is divisible by 8 positive integers
Case b: k = 3, in which case 20k = (20)(3) = 60 = (2^
2)(3^
1)(5^
1). So, the number of divisors = (
2+1)(
1+1)(
1+1) = (3)(2)(2) = 12. In other words,
20k is divisible by 12 positive integers
Since we cannot answer the
target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: k is 7
This means that 20k = (20)(7) = 140
Since we know the EXACT value of 20k, we COULD (very easily) list and count all of positive divisors of 140.
Since we COULD answer the
target question with certainty, statement 2 is SUFFICIENT
Answer =
B
Cheers,
Brent
