Practice GMAT Data Sufficiency Question

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If b is prime and the symbol # represents one of the following operations: addition, subtraction, multiplication, or division, is the value of b # 2 even or odd?

(b # 1) # 2 = 5
4 # b = 3 # (1 # b) and b is even

A)	Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B)	Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C)	BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D)	EACH statement ALONE is sufficient.
E)	Statement (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Correct Answer: D

This problem deals with the properties of prime numbers. Keep in mind that 1 is not a prime number and that 2 is the only even prime number.
Evaluate Statement (1) alone.
1. Try each of the operations in turn. First, try addition:
  (b + 1) + 2 = 5
  Solve for b.
  b = 2.
  Under addition, b = 2, which is a prime number; therefore addition is a possibility for the operator.
2. Next, try subtraction.
  (b - 1) - 2 = 5
  Solve for b.
  b = 8
  But b = 8 is not prime, therefore operator cannot represent subtraction.
3. Next, try multiplication.
  (b * 1) * 2 = 5
  Solve for b.
  b = 5/2
  But b = 5/2 is not prime, therefore operator cannot represent multiplication.
4. Finally, try division.
  (b / 1) / 2 = 5
  Solve for b.
  b = 10
  But b = 10 is not prime, therefore operator cannot represent division.
5. Since addition is the only operation for which b is prime, # must represent addition. In this case, b = 2 and the value of b # 2 is 4, which is even.
6. Statement (1) is SUFFICIENT.
Evaluate Statement (2) alone.
1. Try each of the operations in turn. First, try addition:
  4 + b = 3 + (1 + b)
  Subtract 4 from each side.
  b = b
  While this is true, it does not give any information about the value of b. However, addition is still a possible operation.
2. Next, try subtraction:
  4 - b = 3 - (1 - b)
  Solve for b.
  b = 1
  In this case, b = 1 is not a prime number, so subtraction is not a possible operation.
3. Next, try multiplication:
  4 * b = 3 * (1 * b)
  Simplify.
  4b = 3b
  The only value for which this holds true is b = 0, which is not a prime number. Therefore, multiplication is not a possible operation.
4. Finally, try division:
  4 / b = 3 / (1 / b)
  Multiply both sides by (1 / b)
  4 / b² = 3
  Solve for b².
  b² = 4/3
  Which means that b = sqrt(4/3) or b = -sqrt(4/3). Neither of these is prime, so division is not a possible operation.
5. The symbol # must represent addition, since this is the only possible operation. By Statement (2), b is even, but b is still prime. Since 2 is the only even prime number, b must be 2. In this case, the value of b # 2 is even because an even number plus 2 is still an even number.
6. Statement (2) is SUFFICIENT.
Since Statement (1) alone is SUFFICIENT and Statement (2) alone is SUFFICIENT, answer D is correct.

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