If joshmachine440 is correct (see 3 posts above), then the question
SHOULD read as follows:
If x, y, and z are all positive integers, what is the remainder when 7^(xyz) is divided by 4?
1) yz = 3
2) x is odd.
Target question: What is the remainder when 7^(xyz) is divided by 4
Let's first examine some powers of 7
7¹ = 7 When we divide 7 by 4, the remainder is
3
7² = 49 When we divide 49 by 4, the remainder is
1
7³ = 343 When we divide 343 by 4, the remainder is
3
7� = 2401 When we divide 2401 by 4, the remainder is
1
etc.
So, when 7^(ODD integer) is divided by 4, the remainder is
3
When 7^(EVEN integer) is divided by 4, the remainder is
1
So, the goal here is to determine whether xyz is guaranteed to be either ODD or EVEN.
Statement 1: yz = 3
This does not guarantee that xyz is odd, nor does it guarantee that xyz is even.
Consider these two CONFLICTING cases:
Case a: x = 1, y = 1 and z = 3. Here, xyz = 3, which is ODD. In the case
7^(xyz) divided by 4 leaves remainder 3
Case b: x = 2, y = 1 and z = 3. Here, xyz = 6, which is EVEN. In the case
7^(xyz) divided by 4 leaves remainder 1
Since we cannot answer the
target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: x is odd.
This does not guarantee that xyz is odd, nor does it guarantee that xyz is even.
Consider these two CONFLICTING cases:
Case a: x = 1, y = 1 and z = 1. Here, xyz = 1, which is ODD. In the case
7^(xyz) divided by 4 leaves remainder 3
Case b: x = 1, y = 1 and z = 2. Here, xyz = 2, which is EVEN. In the case
7^(xyz) divided by 4 leaves remainder 1
Since we cannot answer the
target question with certainty, statement 2 is NOT SUFFICIENT
Statements 1 and 2 combined
Statement 1 tells us that yz = 3 = ODD
Statement 2 tells us that x is ODD
So, xyz = (x)(yz) = (ODD)(ODD) = ODD
So, we can be
certain that xyz is ODD, which means
7^(xyz) divided by 4 leaves remainder 3
Since we can answer the
target question with certainty, the combined statements are SUFFICIENT
Answer =
C
Cheers,
Brent
