The positive integer k has exactly two positive prime factors 3 and 7. If k has a total of 6...

Understanding the Question

We need to find the value of k.

Given Information

• k is a positive integer
• k has exactly two prime factors: 3 and 7
• k has exactly 6 positive factors (including 1 and k)

What We Need to Determine

Since \(\mathrm{k = 3^a \times 7^b}\) where a and b are positive integers, we need to find the specific values of a and b. The constraint that k has exactly 6 factors means \(\mathrm{(a+1)(b+1) = 6}\).

Key Insight

With only 2 prime factors and exactly 6 total factors, there are very limited possible structures for k. Since \(\mathrm{6 = 2 \times 3}\), and both a and b must be at least 1, the only possibilities are:

• \(\mathrm{(a+1, b+1) = (2, 3)}\) → meaning \(\mathrm{(a, b) = (1, 2)}\) → \(\mathrm{k = 3^1 \times 7^2 = 3 \times 49 = 147}\)
• \(\mathrm{(a+1, b+1) = (3, 2)}\) → meaning \(\mathrm{(a, b) = (2, 1)}\) → \(\mathrm{k = 3^2 \times 7^1 = 9 \times 7 = 63}\)

So k must be either 147 or 63. To answer the question with certainty, a statement must tell us which one.

Analyzing Statement 1

Statement 1: 9 is a factor of k

Since \(\mathrm{9 = 3^2}\), this means k must contain at least \(\mathrm{3^2}\) in its prime factorization. In other words, the exponent of 3 must be at least 2.

Checking our two possible values:

• If \(\mathrm{k = 147 = 3^1 \times 7^2}\), then 3 appears only to the first power, so 9 is NOT a factor ✗
• If \(\mathrm{k = 63 = 3^2 \times 7^1}\), then \(\mathrm{3^2 = 9}\) IS a factor ✓

Therefore, if 9 is a factor of k, we know definitively that k = 63.

[STOP - Statement 1 is SUFFICIENT!]

This eliminates choices B, C, and E.

Analyzing Statement 2

Forget Statement 1 completely. We're starting fresh.

Statement 2: 49 is a factor of k

Since \(\mathrm{49 = 7^2}\), this means k must contain at least \(\mathrm{7^2}\) in its prime factorization. The exponent of 7 must be at least 2.

Checking our two possible values:

• If \(\mathrm{k = 147 = 3^1 \times 7^2}\), then \(\mathrm{7^2 = 49}\) IS a factor ✓
• If \(\mathrm{k = 63 = 3^2 \times 7^1}\), then 7 appears only to the first power, so 49 is NOT a factor ✗

Therefore, if 49 is a factor of k, we know definitively that k = 147.

[STOP - Statement 2 is SUFFICIENT!]

The Answer: D

Each statement alone uniquely determines the value of k:

• Statement 1 tells us k = 63
• Statement 2 tells us k = 147

Answer Choice D: Each statement alone is sufficient.