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PQP is a three-digit number having digits P and Q;

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PQP is a three-digit number having digits P and Q;

by Jay@ManhattanReview » Fri Jun 09, 2017 5:04 am
stevecultt wrote:PQP is a three-digit number having digits P and Q; and RQS5 is a four-digit number having
digits R, Q, S and 5. What is the value of R?

(1) PQP x P = RQS5
(2) P, Q, R, and S are distinct non-zero digits.

OA A

Need an approach for this problem. Thanks in advance.
Statement 1:

In the multiplication, the unit digit is 5.
Thus, the product P x P has 5 in the unit place ƒ=> P = 5
Thus, the number PQP = 5Q5

ƒ=> PQP x P = 5Q5 x 5

Thus, 5Q5 represents a number between 505 and 595; we can ballpark between 500 and
600.

Since 500 x 5 ƒ= 2500 and 600 x 5 ƒ= 3000, we have: 2500 < RQS5 < 3000

Thus, we have: R = 2. - Sufficient

Statement 2:

We know that P, Q, R and S are distinct non-zero digits.
However, their values cannot be determined. - Insufficient

The correct answer: A

Hope this helps!

Relevant book: Manhattan Review GMAT Data Sufficiency Guide

-Jay
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Last edited by Jay@ManhattanReview on Wed Jul 25, 2018 9:30 pm, edited 1 time in total.
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by [email protected] » Fri Jun 09, 2017 9:40 am
Hi stevecult,

This DS question is all about Number Properties that are tied to specific numbers.

We're told that the DIGITS P, Q, R and S make up the 3-digit number PQP and the 4-digit number RQS5. We're asked for the value of R.

1) (PQP)(P) = RQS5

With Fact 1, we have the product of 2 numbers and the result. Notice how the units digits in the product are both "P"s - with the resulting digit a "5." How many digits can you multiply by themselves and end in a 5? There are only 10 options to consider:
(0)(0) = 0
(1)(1) = 1
(2)(2) = 4
(3)(3) = 9
Etc.

If you map them all out, you'll find that there's just ONE option: 5. This means that the "P" MUST be a 5. Substituting that in, we have...
(5Q5)(5) = RQS5

From here, it looks like the value of Q will impact the value of R, so let's see if we can define the range of possible outcomes by trying a couple of options...
Minimum:
Q=0... (505)(5) = 2525
Maximum:
Q=9... (595)(5) = 2975

Notice that the "R" will ALWAYS be a "2", so there's no more work to be done.
Fact 1 is SUFFICIENT.

2) P, Q, R and S are DISTINCT non-zero digits.

With this Fact, we know that the 4 digits represent 4 DIFFERENT integers, but we don't know exactly what R is.
Fact 2 is INSUFFICIENT

Final Answer: A

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by Matt@VeritasPrep » Thu Jun 22, 2017 11:05 pm
PQP * P = RQS5

Since the right side ends in 5, the left side must also end in 5. This means P is 5. From there,

5Q5 * 5 = RQS5

(500 + 10Q + 5) * 5 = 1000R + 100Q + 10S + 5

2500 + 20 = 1000R + 50Q + 10S

2520 = 1000R + 50Q + 10S

The maximum values for Q and S are each 9, so the maximum of 50Q + 10S = 450 + 90 = 540. That makes R = 1 too small, so R = 2 is the only possibility.
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