W, X, Y, and Z represent distinct digits such that WX * YZ = 1995. Wha

1995 has 4 prime factors 3, 5, 7, 19

and as I've understood task we should make from this 4 numbers two two-digits numbers that give us 1995 as a result of mutiplication
we can multiple 3 * 19 = 57 and 5 * 19 = 95. (7 * 19 will be more than two-digits number)
and second number will be 5 * 7 = 35 and 3 * 7 = 21

we made all prerequisite calculations and for now should look on our statements:

1) x = prime number: we have 57, 95, 35 - insufficient
2) z not prime number: we have only 21 so know we know that YZ = 21 and WX = 95. 21 * 95 = 1995. So W = 9. Sufficient.

Answer B

VERITAS PREP OFFICIAL SOLUTION:

This question can be very tempting to start off with brute force. We can limit our choices by looking at the unit digits. If the unit digit of the product is 5, then there are only a few digits that are possible for X and Z. They all have to be odd, and, more than that, one of them must be exactly 5, as no other digits combine to give a 5. If one of them is 5, the other one is some odd number, 1, 3, 5, 7 or 9. Unfortunately, multiple options exist at both prime (3, 5 and 7) and non-prime (1, 9) for these digits, so it will be hard to narrow down the choices (where’s a dart board when you need one?)

Let’s look at this problem another way, which is: these two numbers must multiply to 1,995. We know one number ends with a 5, so we arbitrarily set it to be 25 and see what that gives if we set the other number to be 91. That comes to 2,275, which is way above what we need. How about 25 * 81, that yields 2,025. That’s too big, but just barely. How about 25 * 79? That will give us 1,975, which is slightly too small. We can’t get 1,995 with 25, but that’s all we’ve demonstrated so far. We can eliminate some choices as number like 15 can never be multiplied by a 2-digit number and yield 1,995, but there are still numerous choices to test.

It’s pretty easy to see how the brute force approach when you have dozens of possibilities will be very tedious. There’s another element that’s even worse, which is let’s say you manage to find a combination that works (such as 21 * 95), how can you be sure that this is the only way to get this product? Short of trying every single possibility (or calling the Psychic Friends hotline), you can’t be sure of your answer.

This problem thus requires a more structured approach, based on mathematical properties and not dumb luck. If two numbers multiply to a specific product, then we can limit the possibilities by using factors. We thus need to factor out 1,995 and we’ll have a much better idea of the limitations of the problem.

1,995 is clearly divisible by 5, but the other number might be hard to produce. The easiest trick here is to think of it as 2,000, and then drop one multiple of 5. Since 2,000 is 5 x 400, this is 5 x 399. Now, 399 is a lot easier than it looks, because it’s clearly divisible by 3 (since the digits add up to 21, which is a multiple of 3). Afterwards, we have 133, which is another tough one, but you might be able to see that it’s divisible by 7, and actually comes to 7 x 19. Finally, since 19 is prime, we have the prime factors of 1,995: 3 x 5 x 7 x 19.

How does this help? Well there may be 16 factors of 1,995, but the limitations of the problem tell us that we only have two two-digit numbers. Thus something like 15 * 133 breaks the rules of the problem. Our only options to avoid 3-digits are 19*3 and 5*7 or 19*5 and 3*7. This gives us either 57 * 35 or 95 * 21. At least at this point we’re 100% sure that these are the only two-digit permutations that combine to give 1,995.

Let’s get back to the problem. Statement 1 tells us that X (the unit digit of the first number) is prime, which knocks out 21 from the running. However the three other options all end with a prime unit digit, meaning that any of them are still possible. At this point it’s very important to note that the problem specified that W, X, Y and Z were all distinct integers. Since they must all be different, the option of 57 * 35 is not valid because the 5 is duplicated. As such, the only option is 95*21, and the prime number restriction confirms that it’s really 95 * 21 (and not 21 * 95). Variable W must be 9, and thus this statement ends up being sufficient.

Statement 2 essentially provides the same information, as Z is not a prime number and thus necessarily 1 given our choices. This confirms that the multiplication is 95 * 21 and W is still 9. Either statement alone is sufficient, so answer choice D is the correct option here. It’s important to note how close this question was to being answer choice B, as the non-prime limitation ensured we knew where the 1 was. But the fact that these digits had to be distinct changed the answer from B to D, reinforcing the adage that you should read the questions carefully.

This question can be solved without factors, but it is very hard to confidently answer it using only a brute-force approach. Solving through mathematics and number properties is not always the easiest route to success on data sufficiency. Sometimes you can write down a few options and see exactly how the problem will unfold, but if you use concrete concepts, you’ll know when it’s been enough.

Given WX * YZ = 1995

With Statement (1), knowing X is prime does not help us determine W. We know that either X or Z has to be 5, but we dont know which one.

With Statement (2), we are told Z is not prime. This means X has to be 5, which is prime (Note that we don't need to know statement 1 here).

Prime factors of 1995 are 3, 5, 7 and 19.

In order for our product to have a 5 in the units digit, we need an odd number to multiply by 5.

If Z is not prime, then it can be either 1 or 9 (rest of the odd integers are prime)

We know so far: W5 * Y(1 or 9) = 1995

Two possibilities: If X is 9, then YZ = 19 (one of the prime factors). But we would need WX to be 105 (3*5*7), which does not make sense.

The other option is that X is 1, which gives us YZ = 21 (3 * 7). Now WX is 95 (5*19) and this satisfies our two digit constraint. Thus W is 9.

Answer is B (Statement 2 alone)

Please let me know if I committed an error in my thinking.

Hi..

factorize 1995..
\(1995=3*5*7*19\)
now 19 has to be multiplied with another prime factor otherwise the other integer becomes 3 digit integer 105.. 19*105
so possiblities--
1)19*3 and 5*7 that is 57 and 35... NOT possible as 5 is repeated
2) 19*5 and 3*7 that is 95 and 21.. possible
3) 19*7 and 3*5... NOT possible as 19*7 becomes 3-digit integer

WX can be 95 or 21..
so W can be 9 or 2

lets see the statements..
(1) X is a prime number
If X is prime WX has to be 95... so W is 9
suff

(2) Z is not a prime number
If Z is not prime, YZ must be 21.
so WZ is 95 and W is 9
suff

D

D is the Answer

factorization of 1995 yields 5x3x7x19 of which 95x21 is the combination we are interested in.

95x21=1995
the distinct digits are 1,2,9,5
w9 x5 y2 z1

1) X is prime = 95x21=1995 , w=9 , hence A or D. ( where X=5)
2) Z is not prime= 95x21=1995, w=9 hence Only D (where Z = 1) ( Therefore A is rejected)

We would like to start from factorizing 1995 to see our options:

1995 = 5 * 399 = 5 * 3 * 133 = 5 * 3 * 7 * 19. (The 133 is a bit tricky but we can't have 133 be a prime number since we need 2 double-digit numbers).

19 is a big number so we should start breaking 1995 with multiples of 19.

19 * (5*3*7) = 19 * 105, 105 too big so not an option.

(19*3) * (5*7) = 57 * 35, note the 5 repeats while the digits should be distinct, so this is also not an option.

(19*5) * (3*7) = 95 * 21. This is our only option to break down the numbers accordingly!

We do not have any more options as (19*7) is already too big. Finally this means we have 95*21 or 21*95, and W is either 2 or 9.

Statement 1:
X is prime, so WX must be 95. Sufficient.

Statement 2:
Z is not prime, YZ must be 21 so WX is 95. Sufficient.

Ans: D

Why we aren't considering -ve digits?

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:

1. (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.
2. (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.
3. (15, 133) - 133 is not a two-digit number.
4. (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

Yeah, I agree with u.
I also think that the answer should be D