If ntext{ and }t are positive integers, what is the greatest prime factor of nt? The greatest common factor of...

Understanding the Question

We need to find the greatest prime factor of the product nt, where n and t are positive integers.

What We Need to Determine

To have sufficiency, we must be able to identify exactly one specific value for the greatest prime factor of nt. This means either:

• We can find the exact values of n and t, OR
• We have enough information about the prime factorization to identify the largest prime factor with certainty

Key Insight

The greatest prime factor of a product depends on all the prime factors present in both numbers being multiplied. Even if we don't know the exact values of n and t, we might still be able to determine their greatest prime factor if we have the right constraints.

Analyzing Statement 1

Statement 1: The greatest common factor of n and t is 5.

This means both n and t are divisible by 5 (since 5 is prime). We can write:

• \(\mathrm{n = 5 \times (some\ integer)}\)
• \(\mathrm{t = 5 \times (some\ integer)}\)

But here's what we don't know: what other prime factors might be present in n or t beyond the common factor of 5.

Testing Different Scenarios

Let's explore specific possibilities to see if we get a consistent answer:

Scenario 1: n = 5, t = 5

• \(\mathrm{nt = 25 = 5^2}\)
• Greatest prime factor = 5 ✓

Scenario 2: n = 5, t = 35 = 5 × 7

• \(\mathrm{nt = 175 = 5^2 \times 7}\)
• Greatest prime factor = 7 ✗

Scenario 3: \(\mathrm{n = 15 = 3 \times 5, t = 25 = 5^2}\)

• \(\mathrm{nt = 375 = 3 \times 5^3}\)
• Greatest prime factor = 5 ✓

Notice how different scenarios yield different greatest prime factors (5 in some cases, 7 in others). This shows we cannot determine a unique answer.

[STOP - Not Sufficient!] Statement 1 alone is NOT sufficient.

This eliminates answer choices A and D.

Analyzing Statement 2

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

Statement 2: The least common multiple of n and t is 105.

First, let's find the prime factorization of 105:

\(\mathrm{105 = 3 \times 5 \times 7}\)

Key Insight About LCM

The LCM contains all prime factors that appear in either n or t, raised to their highest powers. This crucial property tells us:

• The only possible prime factors of n and t are 3, 5, and 7
• No prime larger than 7 can divide either n or t (otherwise it would appear in the LCM)
• Since 7 appears in the LCM, it must be a factor of at least one of n or t

Why This Gives Us the Answer

Since:

1. The number 7 is present in the LCM, and
2. 7 is the largest prime that could possibly divide n or t

We can conclude with certainty that 7 is the greatest prime factor of nt.

It doesn't matter how n and t split up the factors 3, 5, and 7 between them—the product nt will contain all these primes, and 7 will always be the largest.

[STOP - Sufficient!] Statement 2 alone is sufficient.

This eliminates answer choices C and E.

The Answer: B

Statement 2 alone tells us the LCM is \(\mathrm{105 = 3 \times 5 \times 7}\), which guarantees that 7 is the greatest prime factor of nt. Statement 1 alone leaves multiple possibilities open.

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