Bunuel
W, X, Y and Z represent distinct integers such that WX * YZ = 1,995. What is the
block blast value of W?
WX
* YZ
_____
1,995
(1) X is a prime number
(2) Z is not a prime number
We are given that W, X, Y, and Z are distinct integers. The equation is WX * YZ = 1,995. Here, WX and YZ represent two-digit numbers, where W and Y are the tens digits, and X and Z are the units digits. So, the equation can be written as (10W + X) * (10Y + Z) = 1995.
First, let's find the prime factorization of 1995. 1995=5×399 399=3×133 133=7×19 So, 1995=3×5×7×19.
We need to find two two-digit numbers whose product is 1995. Let the two two-digit numbers be A and B, so A * B = 1995.
Let's list the factors of 1995: 1, 3, 5, 7, 15, 19, 21, 35, 57, 95, 105, 133, 285, 399, 665, 1995.
We are looking for pairs of two-digit numbers from this list. Possible pairs (A, B) such that A * B = 1995 and A and B are two-digit numbers:
- (21, 95)
- If WX = 21, then W = 2, X = 1.
- If YZ = 95, then Y = 9, Z = 5.
- Integers are W=2, X=1, Y=9, Z=5. These are distinct. This is a possible solution.
- (35, 57)
- If WX = 35, then W = 3, X = 5.
- If YZ = 57, then Y = 5, Z = 7.
- Integers are W=3, X=5, Y=5, Z=7. Here, X=5 and Y=5 are not distinct. So this is not a valid solution.
- (15, 133) - 133 is not a two-digit number.
- (7, 285) - 7 is not a two-digit number.
So far, the only valid pair of two-digit numbers that leads to distinct W, X, Y, Z is (21, 95). Let's consider the permutations: Case 1: WX = 21 and YZ = 95 W = 2, X = 1, Y = 9, Z = 5. Are W, X, Y, Z distinct? Yes (2, 1, 9, 5 are all different). Case 2: WX = 95 and YZ = 21 W = 9, X = 5, Y = 2, Z = 1. Are W, X, Y, Z distinct? Yes (9, 5, 2, 1 are all different).
So, from the problem statement alone, W could be 2 or 9. We need to use the statements.
Statement (1): X is a prime number. From Case 1: WX = 21, so X = 1. 1 is not a prime number. So Case 1 is eliminated by Statement (1). From Case 2: WX = 95, so X = 5. 5 is a prime number. So Case 2 is consistent with Statement (1). If Case 2 is the only possibility, then W = 9.
Let's check if there are other pairs of two-digit numbers. The factors of 1995 are 3, 5, 7, 19. Possible combinations for two two-digit numbers:
- (3×7)×(5×19)=21×95
- If WX = 21, X = 1 (not prime).
- If WX = 95, X = 5 (prime). This gives W = 9. W,X,Y,Z are 9,5,2,1. All distinct.
- (3×5)×(7×19)=15×133 (133 is not a two-digit number)
- (3×19)×(5×7)=57×35
- If WX = 57, X = 7 (prime). This gives W = 5. YZ = 35, Y = 3, Z = 5. W=5, X=7, Y=3, Z=5. Here X=7 and Z=5 are prime. But W=5 and Z=5 are not distinct. So this solution is invalid.
- If WX = 35, X = 5 (prime). This gives W = 3. YZ = 57, Y = 5, Z = 7. W=3, X=5, Y=5, Z=7. Here X=5 and Y=5 are not distinct. So this solution is invalid.
So, with Statement (1), the only remaining valid case is WX = 95 and YZ = 21. This implies W = 9, X = 5, Y = 2, Z = 1. All are distinct integers: 9, 5, 2, 1. X = 5, which is prime. So, Statement (1) is sufficient to determine W.
Statement (2): Z is not a prime number. Let's revisit our two initial valid cases without considering Statement (1): Case 1: WX = 21 and YZ = 95 W = 2, X = 1, Y = 9, Z = 5. Is Z = 5 not a prime number? No, 5 is a prime number. So Case 1 is eliminated by Statement (2). Case 2: WX = 95 and YZ = 21 W = 9, X = 5, Y = 2, Z = 1. Is Z = 1 not a prime number? Yes, 1 is not a prime number. So Case 2 is consistent with Statement (2). If Case 2 is the only possibility, then W = 9.
Let's re-examine all possible valid combinations of W, X, Y, Z for being distinct and being two-digit numbers. The pairs of two-digit numbers whose product is 1995 are (21, 95) and (35, 57).
Pair 1: (21, 95) a) WX = 21 => W = 2, X = 1. YZ = 95 => Y = 9, Z = 5. Distinct: (2, 1, 9, 5) - Yes. Apply Statement (2): Z = 5. Is Z not a prime number? No, 5 is prime. So this case is eliminated. b) WX = 95 => W = 9, X = 5. YZ = 21 => Y = 2, Z = 1. Distinct: (9, 5, 2, 1) - Yes. Apply Statement (2): Z = 1. Is Z not a prime number? Yes, 1 is not prime. So this case is valid. W = 9.
Pair 2: (35, 57) a) WX = 35 => W = 3, X = 5. YZ = 57 => Y = 5, Z = 7. Distinct: (3, 5, 5, 7) - No, X=Y=5. This case is invalid regardless of Statement (2). b) WX = 57 => W = 5, X = 7. YZ = 35 => Y = 3, Z = 5. Distinct: (5, 7, 3, 5) - No, W=Z=5. This case is invalid regardless of Statement (2).
So, with Statement (2), the only remaining valid case is WX = 95 and YZ = 21. This implies W = 9. So, Statement (2) is sufficient to determine W.
Since both statements individually are sufficient, the answer is D.
The final answer is D