If p and n are positive integers and p > n, what is the remainder when p^2 - n^2 is...

Understanding the Question

We need to find the remainder when \(\mathrm{p}^2 - \mathrm{n}^2\) is divided by 15, where p and n are positive integers with \(\mathrm{p} > \mathrm{n}\).

What We Need to Determine

Using the identity \(\mathrm{p}^2 - \mathrm{n}^2 = (\mathrm{p} + \mathrm{n})(\mathrm{p} - \mathrm{n})\), we're looking for the remainder when \((\mathrm{p} + \mathrm{n})(\mathrm{p} - \mathrm{n})\) is divided by 15.

Here's the key insight: Since \(15 = 3 \times 5\), we need to know the remainder when \((\mathrm{p} + \mathrm{n})(\mathrm{p} - \mathrm{n})\) is divided by both 3 and 5. Think of it like needing both coordinates to locate a point - we need complete information about both factors modulo both prime divisors.

Definition of Sufficiency

For this value question to be sufficient, we must be able to determine exactly ONE remainder when \(\mathrm{p}^2 - \mathrm{n}^2\) is divided by 15. If different valid pairs of (p, n) can produce different remainders, then we don't have sufficiency.

Analyzing Statement 1

Statement 1 tells us that when p + n is divided by 5, the remainder is 1.

What This Means

We know (p + n) leaves remainder 1 when divided by 5. But remember, \(\mathrm{p}^2 - \mathrm{n}^2 = (\mathrm{p} + \mathrm{n})(\mathrm{p} - \mathrm{n})\), so we need information about BOTH factors.

What We're Missing

• We don't know anything about (p - n) modulo 5
• We don't know anything about either factor modulo 3

Testing the Logic

Many different pairs can have \(\mathrm{p} + \mathrm{n} \equiv 1 \pmod{5}\). For instance:

• p = 6, n = 5: sum is 11 (remainder 1 when divided by 5)
• p = 8, n = 3: sum is 11 (remainder 1 when divided by 5)

But these pairs give different values of \(\mathrm{p}^2 - \mathrm{n}^2\):

• First pair: 36 - 25 = 11 → remainder 11 when divided by 15
• Second pair: 64 - 9 = 55 → remainder 10 when divided by 15

Since we can get different remainders from valid pairs, Statement 1 alone is NOT sufficient.

[STOP - 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 tells us that when p - n is divided by 3, the remainder is 1.

What This Provides

We know (p - n) leaves remainder 1 when divided by 3. This gives us partial information about one factor.

What's Still Unknown

• We don't know anything about (p + n) modulo 3
• We don't know anything about either factor modulo 5

Logical Analysis

Just as with Statement 1, having information about only one factor (and only modulo one prime) isn't enough. Many pairs can satisfy \(\mathrm{p} - \mathrm{n} \equiv 1 \pmod{3}\):

• p = 4, n = 3: difference is 1
• p = 7, n = 6: difference is 1

These yield different values of \(\mathrm{p}^2 - \mathrm{n}^2\) with potentially different remainders modulo 15.

Statement 2 alone is NOT sufficient.

[STOP - Not Sufficient!] This eliminates choice B (we already eliminated D).

Combining Statements

Now let's see what happens when we use both statements together.

Combined Information

From both statements:

• \(\mathrm{p} + \mathrm{n} \equiv 1 \pmod{5}\)
• \(\mathrm{p} - \mathrm{n} \equiv 1 \pmod{3}\)

The Critical Gap

Even with both statements, we still have incomplete information:

• We know (p + n) mod 5, but not mod 3
• We know (p - n) mod 3, but not mod 5

It's like having partial coordinates - we need complete information about both factors modulo both 3 and 5 to determine the remainder modulo 15.

Quick Verification

Consider pairs where p - n = 1 (consecutive integers, which automatically satisfy Statement 2). Many such pairs can also have \(\mathrm{p} + \mathrm{n} \equiv 1 \pmod{5}\):

• (6, 5): sum = 11 ≡ 1 (mod 5), and \(\mathrm{p}^2 - \mathrm{n}^2 = 36 - 25 = 11\)
• (11, 10): sum = 21 ≡ 1 (mod 5), and \(\mathrm{p}^2 - \mathrm{n}^2 = 121 - 100 = 21\)
• (16, 15): sum = 31 ≡ 1 (mod 5), and \(\mathrm{p}^2 - \mathrm{n}^2 = 256 - 225 = 31\)

These values (11, 21, 31) have different remainders when divided by 15:

• 11 ÷ 15 = 0 remainder 11
• 21 ÷ 15 = 1 remainder 6
• 31 ÷ 15 = 2 remainder 1

Even both statements together are NOT sufficient.

[STOP - Not Sufficient!] This eliminates choice C.

The Answer: E

The statements together are not sufficient because we still lack complete information about the remainder when \(\mathrm{p}^2 - \mathrm{n}^2\) is divided by 15.

Answer Choice E: "The statements together are not sufficient."