Loading...
If \(\mathrm{p}\) and \(\mathrm{n}\) are positive integers and \(\mathrm{p} > \mathrm{n}\), what is the remainder when \(\mathrm{p}^2 - \mathrm{n}^2\) is divided by 15?
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 \(\mathrm{p} > \mathrm{n}\).
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.
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.
Statement 1 tells us that when p + n is divided by 5, the remainder is 1.
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.
Many different pairs can have \(\mathrm{p} + \mathrm{n} \equiv 1 \pmod{5}\). For instance:
But these pairs give different values of \(\mathrm{p}^2 - \mathrm{n}^2\):
Since we can get different remainders from valid pairs, Statement 1 alone is NOT sufficient.
[STOP - Not Sufficient!] This eliminates choices A and D.
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.
We know (p - n) leaves remainder 1 when divided by 3. This gives us partial information about one factor.
Just as with Statement 1, having information about only one factor (and only modulo one prime) isn't enough. Many pairs can satisfy \(\mathrm{p} - \mathrm{n} \equiv 1 \pmod{3}\):
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).
Now let's see what happens when we use both statements together.
From both statements:
Even with both statements, we still have incomplete information:
It's like having partial coordinates - we need complete information about both factors modulo both 3 and 5 to determine the remainder modulo 15.
Consider pairs where p - n = 1 (consecutive integers, which automatically satisfy Statement 2). Many such pairs can also have \(\mathrm{p} + \mathrm{n} \equiv 1 \pmod{5}\):
These values (11, 21, 31) have different remainders when divided by 15:
Even both statements together are NOT sufficient.
[STOP - Not Sufficient!] This eliminates choice C.
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."