If m is a positive odd integer between 2 and 30, then m is divisible by how many different positive...

Understanding the Question

Let's break down what we're looking for: How many different positive prime numbers divide m?

Given Information

\(\mathrm{m}\) is a positive odd integer
\(2 < \mathrm{m} < 30\)
Since \(\mathrm{m}\) is odd, we know \(\mathrm{m}\) is NOT divisible by 2


What We Need to Determine
We need to find the exact count of different prime numbers that divide \(\mathrm{m}\). This is a specific value question - we need one definite number.

Key Insights
Since \(\mathrm{m}\) is odd, it cannot be divisible by 2. This means we're only considering odd primes as potential divisors:

Odd primes less than 30: 3, 5, 7, 11, 13, 17, 19, 23, 29
The possible values of \(\mathrm{m}\) are: 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29


The number of prime divisors varies greatly depending on which odd integer \(\mathrm{m}\) is. For example:

If \(\mathrm{m} = 7\) (prime), it has exactly 1 prime divisor: just 7
If \(\mathrm{m} = 15 = 3 \times 5\), it has exactly 2 prime divisors: 3 and 5
If \(\mathrm{m} = 9 = 3^2\), it has exactly 1 prime divisor: just 3


Analyzing Statement 1

Statement 1: m is not divisible by 3.

What Statement 1 Tells Us
This eliminates all multiples of 3 from our possible values of \(\mathrm{m}\). So \(\mathrm{m}\) cannot be: 3, 9, 15, 21, or 27.

The remaining possibilities are: 5, 7, 11, 13, 17, 19, 23, 25, 29.

Testing Different Scenarios
Let's examine what types of numbers remain:

Prime numbers: 5, 7, 11, 13, 17, 19, 23, 29
Non-prime number: \(25 = 5^2\)


Here's the crucial insight: All remaining values except 25 are prime numbers, and each prime number has exactly 1 prime divisor (itself). Even \(25 = 5^2\) has exactly 1 prime divisor (just 5).

Verification
Let's double-check by testing each value:

\(\mathrm{m} = 5\): prime divisor is 5 → count = 1
\(\mathrm{m} = 7\): prime divisor is 7 → count = 1
\(\mathrm{m} = 11\): prime divisor is 11 → count = 1
\(\mathrm{m} = 25 = 5^2\): prime divisor is 5 → count = 1
(All other values are prime → count = 1)


Conclusion
Since every possible value of \(\mathrm{m}\) has exactly 1 prime divisor, we can definitively answer the question.

[STOP - Statement 1 is SUFFICIENT!]

This eliminates choices B, C, and E.

Analyzing Statement 2

Now let's forget Statement 1 completely and analyze Statement 2 independently.

Statement 2: m is not divisible by 5.

What Statement 2 Provides
This eliminates all multiples of 5 from our possible values. So \(\mathrm{m}\) cannot be: 5, 15, or 25.

The remaining possibilities are: 3, 7, 9, 11, 13, 17, 19, 21, 23, 27, 29.

Testing Different Scenarios
Let's examine several of these values to see if we get a consistent answer:

If \(\mathrm{m} = 3\) (prime): Has exactly 1 prime divisor (3)
If \(\mathrm{m} = 7\) (prime): Has exactly 1 prime divisor (7)
If \(\mathrm{m} = 9 = 3^2\): Has exactly 1 prime divisor (3)
If \(\mathrm{m} = 21 = 3 \times 7\): Has exactly 2 prime divisors (3 and 7)


We found different results! Some values give us 1 prime divisor, while others give us 2 prime divisors.

Conclusion
Statement 2 is NOT sufficient because different values of \(\mathrm{m}\) lead to different numbers of prime divisors.

This eliminates choices B and D.

The Answer: A

Statement 1 alone gives us a definitive answer (exactly 1 prime divisor for all possible values of \(\mathrm{m}\)), while Statement 2 alone does not provide a unique answer.

Answer Choice A: "Statement 1 alone is sufficient, but Statement 2 alone is not sufficient."