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If N is a positive two-digit integer, is N+1 prime?

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If N is a positive two-digit integer, is N+1 prime?

by fskilnik@GMATH » Thu Mar 14, 2019 6:44 am

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GMATH practice exercise (Quant Class 16)

If N is a positive two-digit integer, is N+1 prime?

(1) The sum of the digits of N is equal to 11.
(2) N-1 is divisible by 7.

Answer: [spoiler]_____(A)__[/spoiler]
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Source: — Data Sufficiency |
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Is N+1 prime?

by GMATGuruNY » Thu Mar 14, 2019 7:28 am

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fskilnik@GMATH wrote:GMATH practice exercise (Quant Class 16)

If N is a positive two-digit integer, is N+1 prime?

(1) The sum of the digits of N is equal to 11.
(2) N-1 is divisible by 7.


Statement 1:
Any integer whose digits sum to a multiple of 3 must itself be a multiple of 3.
Since N has a digit sum of 11, N+1 must have a digit sum of 12.
One exception: If N=29, then N+1 = 30, which has a digit sum of 3.
Since the digit sum of N+1 must be a multiple of 3 -- either 12 or 3 -- N+1 itself must be a 2-digit multiple of 3 and thus cannot be prime.
Since N+1 is not prime, the answer to the question stem is NO.
SUFFICIENT.

Statement 2:
N-1 = 14, 21, 28...
N = 15, 22, 29...
N+1 = 16, 23, 30...
If N+1 = 16, the answer to the question stem is NO.
If N+1 = 23, the answer to the question stem is YES.
INSUFFICIENT.

The correct answer is A.
Last edited by GMATGuruNY on Thu Mar 14, 2019 9:07 am, edited 2 times in total.
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by Ian Stewart » Thu Mar 14, 2019 8:30 am

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GMATGuruNY wrote: Since N has a digit sum of 11, N+1 must have a digit sum of 12.
The logic of your answer is perfect, but this is not quite true - when you add 1 to a number, when that number ends in 9, the digit sum drops by 8 (the tens digit goes up by 1, the units digit falls by 8) rather than increases by 1. So here, when N = 29, after adding 1 we don't get a digit sum of 12, but instead of 3. That's still divisible by 3 of course.
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by GMATGuruNY » Thu Mar 14, 2019 9:02 am

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Ian Stewart wrote:The logic of your answer is perfect, but this is not quite true - when you add 1 to a number, when that number ends in 9, the digit sum drops by 8 (the tens digit goes up by 1, the units digit falls by 8) rather than increases by 1. So here, when N = 29, after adding 1 we don't get a digit sum of 12, but instead of 3. That's still divisible by 3 of course.
Good catch.
In my solution, the lone exception (N=29) is now noted.
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If N is a positive two-digit integer, is N+1 prime?

by fskilnik@GMATH » Thu Mar 14, 2019 12:43 pm

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fskilnik@GMATH wrote:GMATH practice exercise (Quant Class 16)

If N is a positive two-digit integer, is N+1 prime?

(1) The sum of the digits of N is equal to 11.
(2) N-1 is divisible by 7.
Thank you both (Mitch and Ian) for your nice contributions!

$$N = \left\langle {AB} \right\rangle $$
$$N + 1\,\,\mathop = \limits^? \,\,{\rm{prime}}$$
$$\left( 1 \right)\,\,A + B = 11\,\,\,\,\left\{ \matrix{
\,B\,\,{\rm{odd}}\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\,\,\,\,\,\left[ {N + 1\,\,{\rm{even}}\,\, > 2\,\,} \right] \hfill \cr
\,B\,\,{\rm{even}}\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\,\,\,\,\,\left[ {N + 1\,\, \in \left\{ {92 + 1,74 + 1,56 + 1,38 + 1} \right\}\,\,\, \Rightarrow \,\,\,{\rm{multiple}}\,\,{\rm{of}}\,\,3\,\,\, > 3} \right]\,\, \hfill \cr} \right.$$
$$\left( 2 \right)\,\,{{N - 1} \over 7} = {\mathop{\rm int}} \,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,N = 15\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr
\,{\rm{Take}}\,\,N = 22\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr} \right.$$

The correct answer is (A).


We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.

P.S. (to the careful reader): there is at least one number N that satisfies the question stem (pre-statements) and both statements together (92). This is expected to avoid "internal contradictions".
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