What is the value of the positive integer m? When m is divided by 6, the remainder is 3. When...

Understanding the Question

We need to find the value of the positive integer m. This is a value question - we need to determine exactly one specific value for m.

Given information:

• m is a positive integer

What constitutes sufficiency: We need information that narrows down m to exactly one positive integer value. If we can determine multiple possible values or cannot determine any specific value, the information is not sufficient.

Analyzing Statement 1

Statement 1: When m is divided by 6, the remainder is 3.

This tells us that m follows a specific pattern. When we divide by 6 and get remainder 3, m could be:

• 3 (since \(3 \div 6 = 0\) remainder \(3\))
• 9 (since \(9 \div 6 = 1\) remainder \(3\))
• 15 (since \(15 \div 6 = 2\) remainder \(3\))
• 21 (since \(21 \div 6 = 3\) remainder \(3\))
• And so on...

We can see that m could be any number of the form \(6\mathrm{k} + 3\) where k is a non-negative integer (0, 1, 2, 3...). Since there are infinitely many such values, we cannot determine a unique value for m.

[STOP - Not Sufficient!]

Statement 1 alone is NOT sufficient.

This eliminates choices A and D.

Analyzing Statement 2

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

Statement 2: When 15 is divided by m, the remainder is 6.

Here's the key insight: For a division to have remainder 6, the divisor (m) must be greater than 6. Why? Because the remainder must always be less than the divisor - otherwise, you could divide at least one more time!

So we know immediately: \(\mathrm{m} > 6\)

Now, if 15 divided by m leaves remainder 6, that means m divides evenly into \((15 - 6) = 9\).

In other words: \(15 = \mathrm{m} \times \text{(some whole number)} + 6\)

This means: \(\mathrm{m} \times \text{(some whole number)} = 9\)

Let's think about which positive integers greater than 6 divide 9:

• The divisors of 9 are: 1, 3, and 9
• But we need \(\mathrm{m} > 6\)
• The only divisor of 9 that's greater than 6 is 9 itself!

Let's verify: \(15 \div 9 = 1\) with remainder \(6\) ✓

Therefore, m must equal 9. This gives us exactly one value.

[STOP - Sufficient!]

Statement 2 alone is sufficient.

This eliminates choices A, C, and E.

The Answer: B

Statement 2 alone provides enough information to determine that \(\mathrm{m} = 9\), while Statement 1 alone gives us infinitely many possibilities.

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