sivaelectric wrote:1. If sqrt x is a positive integer, is sqrt x a prime number?
- A. x is divisible by exactly 3 positive numbers.
B. All positive factors of x are odd.
Statement 1: x is divisible by exactly 3 positive numbers.
To determine the number of factors of an integer:
1. Prime-factorize the integer
2. Add 1 to each exponent
3. Multiply
For example:
24 = 2^3 * 3^1.
Adding 1 to each exponent and multiplying, we determine that 24 has (3+1)*(1+1)= 8 factors.
Thus, in order for x to have exactly 3 positive factors, x = (prime number)^2. Adding 1 to the exponent, we can see that x will have exactly 2+1=3 factors.
Since x is the square of a prime number, √x is prime.
Sufficient.
Statement 2: All positive factors of x are odd.
The factors of x=9 are 1,3,9. √9 = 3, which is prime.
The factors of x=81 are 1,3,9,27,81. √81=9, which is not prime.
Insufficient.
The correct answer is
A.
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