In the decimal representation of x, where 0 , is the tenths digit of (x nonzero? 16x is an integer....

Understanding the Question

We need to determine whether the tenths digit of \(\mathrm{x}\) is nonzero, where \(\mathrm{x}\) is a decimal between 0 and 1.

Since \(0 < \mathrm{x} < 1\), we can write \(\mathrm{x}\) as \(0.\mathrm{abcd}\ldots\) where 'a' is the tenths digit. The question asks: Is \(\mathrm{a} \neq 0\)?

This is equivalent to asking: Is \(\mathrm{x} \geq 0.1\)?

• If \(\mathrm{x} \geq 0.1\), then the tenths digit must be nonzero (1, 2, 3, etc.)
• If \(\mathrm{x} < 0.1\), then the tenths digit is 0

To have sufficiency for this yes/no question, we need a definitive answer about whether \(\mathrm{x} \geq 0.1\).

Analyzing Statement 1

Statement 1: \(16\mathrm{x}\) is an integer.

Since \(0 < \mathrm{x} < 1\) and \(16\mathrm{x}\) must be an integer, we know that \(\mathrm{x} = \frac{\mathrm{k}}{16}\) for some integer \(\mathrm{k}\) where \(1 \leq \mathrm{k} \leq 15\).

The key insight: Whether \(\frac{\mathrm{k}}{16}\) has a nonzero tenths digit depends on the specific value of \(\mathrm{k}\). Let's test critical cases:

• If \(\mathrm{k} = 1\): \(\mathrm{x} = \frac{1}{16} = 0.0625\) → tenths digit is 0 (the answer is NO)
• If \(\mathrm{k} = 2\): \(\mathrm{x} = \frac{2}{16} = \frac{1}{8} = 0.125\) → tenths digit is 1 (the answer is YES)

Since we get different answers depending on the value of \(\mathrm{k}\), Statement 1 is NOT sufficient.

This eliminates choices A and D.

Analyzing Statement 2

Now let's analyze Statement 2 independently, forgetting Statement 1 completely.

Statement 2: \(8\mathrm{x}\) is an integer.

Since \(0 < \mathrm{x} < 1\) and \(8\mathrm{x}\) must be an integer, we know that \(\mathrm{x} = \frac{\mathrm{k}}{8}\) for some integer \(\mathrm{k}\) where \(1 \leq \mathrm{k} \leq 7\).

Here's the crucial insight: The smallest possible value of \(\mathrm{x}\) is when \(\mathrm{k} = 1\), giving us \(\mathrm{x} = \frac{1}{8} = 0.125\).

Since \(0.125 > 0.1\), this already has a nonzero tenths digit (specifically, 1).

All other possible values are even larger:

• \(\mathrm{k} = 2\): \(\mathrm{x} = \frac{2}{8} = 0.25\)
• \(\mathrm{k} = 3\): \(\mathrm{x} = \frac{3}{8} = 0.375\)
• And so on...

Each of these exceeds \(0.1\) and therefore has a nonzero tenths digit.

Therefore, Statement 2 guarantees that \(\mathrm{x} \geq 0.125 > 0.1\), ensuring the tenths digit is always nonzero.

[STOP - Sufficient!] Statement 2 is sufficient.

This eliminates choices C and E.

The Answer: B

Statement 2 alone is sufficient because it guarantees \(\mathrm{x} \geq \frac{1}{8} = 0.125\), which always has a nonzero tenths digit.

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