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gmatophobia
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Difficulty:
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Question Stats:
69% (02:38) correct
31%
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Let p and q each represent a digit from 0 through 9 and let pq represent a 2-digit positive prime number such that \(20 < pq < 99\). For example, if \(p = 2\) and \(q = 3\), then the 2-digit prime number \(pq\) is 23. If the 2-digit integer \(qp\) is 27 greater than \(pq\), what is the sum of the digits p and q?
A. \(5\)
B. \(7\)
C. \(9\)
D. \(11\)
E. \(13\)
Screenshot 2023-12-10 004311.png [ 126.5 KiB | Viewed 22903 times ]
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A. \(5\)
B. \(7\)
C. \(9\)
D. \(11\)
E. \(13\)
Attachment:
Screenshot 2023-12-10 004311.png [ 126.5 KiB | Viewed 22903 times ]
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Power of Tiny Gains
1.01^(365) = 37.78
0.99^(365) = 0.03
Power of Tiny Gains
1.01^(365) = 37.78
0.99^(365) = 0.03
GMAT 1: 800 Q51 V51
Posts: 2,162
Let \(p\) and \(q\) each represent a digit from \(0\) through \(9\) and let \(pq\) represent a \(2\)-digit positive prime number such that \(20 < pq < 99\). For example, if \(p = 2\) and \(q = 3\), then the \(2\)-digit prime number \(pq\) is \(23\). If the \(2\)-digit integer \(qp\) is \(27\) greater than \(pq\), what is the sum of the digits \(p\) and \(q\)?
The value of \(2\)-digit number \(qp\) is \(10q + p\).
The value of \(2\)-digit number \(pq\) is \(10p + q\).
So, if \(qp - pq = 27\), then \((10q + p) - (10p + q) = 9q - 9p = 27\). Thus, \(q - p = 3\).
Since \(q\) must be \(3\) greater than \(p\), the only possibilities for \(pq > 20\) are \(25\), \(36\), \(47\), \(58\), and \(69\).
\(25\), \(36\), \(58\), and \(69\) are not prime. So, \(47\) is the only possible \(pq\). We can confirm that \(74 - 47 = 27\).
Thus the sum of \(p\) and \(q\) is \(4 + 7 = 11\).
Correct Answer
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The value of \(2\)-digit number \(qp\) is \(10q + p\).
The value of \(2\)-digit number \(pq\) is \(10p + q\).
So, if \(qp - pq = 27\), then \((10q + p) - (10p + q) = 9q - 9p = 27\). Thus, \(q - p = 3\).
Since \(q\) must be \(3\) greater than \(p\), the only possibilities for \(pq > 20\) are \(25\), \(36\), \(47\), \(58\), and \(69\).
\(25\), \(36\), \(58\), and \(69\) are not prime. So, \(47\) is the only possible \(pq\). We can confirm that \(74 - 47 = 27\).
Thus the sum of \(p\) and \(q\) is \(4 + 7 = 11\).
Correct Answer
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General Discussion
RahulJain293
Joined: 24 Apr 2022
Last visit: 13 Sep 2026
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GMAT Focus 1: 585 Q81 V80 DI76 
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MartyMurray
I like this question! Got it wrong in my mock but found interesting way to solve it...
Since we know q = p +3, we can narrow down possible options of p & q based on the scope and question stem
p can be 2,3,4,5,6 (NOT 7,8,9 since it has to smaller than Q by 3)
q out of 5,6,7,8,9 CAN ONLY be 7 & 9 since it is UNIT's digit and prime..
Hence we can quickly make all possible combos and eliminate to find that ONLY 47 is where the difference is 3 between the digits and we can also double check by adding 27 to 47 to get 74 (swapped digits)
Hence 11
Archit3110
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given
10q+p- 10p-q = 27
q-p = 3
possible number such that pq is prime ; 47
sum is 11
option D
10q+p- 10p-q = 27
q-p = 3
possible number such that pq is prime ; 47
sum is 11
option D
gmatophobia
gmatophobia
can any one explain to me why A is wrong ?
41 is a prime no and when we interchange the digits then we got 14 and according to question
(41-14=27) then the some of p+q is 5 how it is wrong?
Archit3110
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Jake7Wimmer
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Last visit: 24 Jan 2025
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GMAT Focus 1: 625 Q79 V83 DI81
GMAT 1: 650 Q48 V32
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I actually think it's just easy to list out the primes starting with 23 and flipping them to see your options. Doesn't take too long because they're aren't actually that many.
After listing out a few you're kinda just left with 41 -> 14 and 47 -> 74.
If you read the question, the lather number needs to be larger so 47 -> 74 is the only answer and adding the sums = 11.
Really only takes like 30 seconds this question.
After listing out a few you're kinda just left with 41 -> 14 and 47 -> 74.
If you read the question, the lather number needs to be larger so 47 -> 74 is the only answer and adding the sums = 11.
Really only takes like 30 seconds this question.
Posts: 1
why is 63-36=27 is wrong? it satisfies the equation. some questions just make me sick!
firuzaadilzadeh
We are told that the smaller number, pq, is prime, whereas 36 in your example is not. You should read questions more carefully.
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gmatophobia
We are given that qp - pq = 27
\((10q + p) - (10p + q) = 27\)
\(9(q-p) = 27\)
\((q-p) = 3\)
Hence various values of p and q are possible. Neither p nor q can be 0 because both pq and qp are two digit numbers.
p = 1, q = 4, Then pq = 14 (Not prime)
p = 2, q = 5, Then pq = 25 (Not prime)
p = 3, q = 6, Then pq = 36 (Not prime)
p = 4, q = 7, Then pq = 47 (Prime!)
We needed pq to be prime (but not qp necessarily) and when p = 4, q = 7, pq is prime. Here p + q = 11.
Other values of p and q are also possible but we will not evaluate them because this is a PS question and here there will be only one correct answer.
Hence answer (D)
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gmatophobia
Given pq = 23, 29, 31, 39, 41, 43, 47, 53, 57, 59, 61, 67....97
10p + q = 27 + 10q + p
P + q=?
10p + q + 27 = 10q + p
27 = 9q – 9p
27/9 = q-p = 3
Therefore q can be 5,7,9 (Q cannot be even, and q must be greater than 3)
And p can be 2, 4, 6
So possible combinations are, 25, 47, 69
25 and 69 aren’t prime, so 47 is the only option
Lets check qp = 27 + pq, which is 27 + 47 = 74
Therefore, q+p = 7 + 4 =11
KarishmaB
Why did you add the coefficient "10" to the equation?
2-digit integer qp can be written as 10q + p because q is 10's digit and p is units digit. Example, if qp = 74, here q=7 and p=4
10q + p = 10*(7) + 4 = 70 + 4 = 74.
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10q + p = 10*(7) + 4 = 70 + 4 = 74.
Danou
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Danou
Danou
Look there are 2 ways of talking about a two digit number. I can say a two digit number is x. I can say when we reverse its digits, the number becomes y. If the difference between them is 9, I can write it as x - y = 9. But how do I solve it now? I have lost some important information - that digits in y are reverse of digits in x.
So in this case I can say that say the first two digit number is pq (where p is tens digit and q is units digit). Then the second number is qp (where q is tens digit and p is unit digit). Now this captures the info of reversing of the digits. But when I want to now calculate the difference between them, I must write it as
10p + q because the value of pq = 10p+q.
Essentially, when p = 3 and q = 2, pq = 32 = 3*10 + 2
I must take the place value of the digit into account too.
A three digit number per = 100p + 10q + r where p, q and r are the digits of the number (say 8, 5 and 1).
851 = 100*8 + 10*5 + 1
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Let p and q each represent a digit from 0 through 9 and let pq represent a 2-digit positive prime number such that \(20 < pq < 99\). For example, if \(p = 2\) and \(q = 3\), then the 2-digit prime number \(pq\) is 23.
If the 2-digit integer \(qp\) is 27 greater than \(pq\), what is the sum of the digits p and q?
20 < 10p + q < 99
(10q + p) - (10p + q) = 27
9(q-p) = 27
q - p = 3
q = p + 3 < 10
p < 7
20 < 10p + p + 3 < 99
20 < 11p + 3 < 99
17 < 11p < 96
p = {2,3,4,5,6}
p = 2 ; q = p+3 = 5; 25 is NOT a prime number
p = 3 ; q = p+3 = 6; 36 is NOT a prime number
p = 4 ; q = p+3 = 7; 47 is a prime number
p = 5 ; q = p+3 = 8; 58 is NOT a prime number
p = 6 ; q = p+3 = 9; 69 is NOT a prime number
pq = 47
Sum of digits of pq = 4+ 7 = 11
IMO D
If the 2-digit integer \(qp\) is 27 greater than \(pq\), what is the sum of the digits p and q?
20 < 10p + q < 99
(10q + p) - (10p + q) = 27
9(q-p) = 27
q - p = 3
q = p + 3 < 10
p < 7
20 < 10p + p + 3 < 99
20 < 11p + 3 < 99
17 < 11p < 96
p = {2,3,4,5,6}
p = 2 ; q = p+3 = 5; 25 is NOT a prime number
p = 3 ; q = p+3 = 6; 36 is NOT a prime number
p = 4 ; q = p+3 = 7; 47 is a prime number
p = 5 ; q = p+3 = 8; 58 is NOT a prime number
p = 6 ; q = p+3 = 9; 69 is NOT a prime number
pq = 47
Sum of digits of pq = 4+ 7 = 11
IMO D
Posts: 385
Classic digit swap scenario, set up as usual:
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they specified it is a prime number, I did the same mistake.
firuzaadilzadeh
Posts: 2
63 is not prime.
63/3=21.
63/3=21.
firuzaadilzadeh
Posts: 2
exactly!!! what's wrong with 41-14 = 27?
shivanks
darfungmatbro
The order is reversed. The question says qp is 27 greater than pq, so (qp-pq=27). For 41, (14-41=-27), meaning 14 is 27 less than 41.
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