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# Is the prime number q equal to 29 ?

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Math Expert
Joined: 02 Sep 2009
Posts: 58390
Is the prime number q equal to 29 ?  [#permalink]

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25 Dec 2017, 05:42
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Difficulty:

95% (hard)

Question Stats:

40% (02:16) correct 60% (02:19) wrong based on 92 sessions

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Is the prime number q equal to 29 ?

(1) q - 1 has exactly 6 positive factors.
(2) 2 and 3 are prime factors of q + 1

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Re: Is the prime number q equal to 29 ?  [#permalink]

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25 Dec 2017, 05:44
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Re: Is the prime number q equal to 29 ?  [#permalink]

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25 Dec 2017, 08:42
Bunuel wrote:
Is the prime number q equal to 29 ?

(1) q - 1 has exactly 6 positive factors.
(2) 2 and 3 are prime factors of q + 1

i. Prime Numbers q with exactly 6 factors of q-1 are: 29, 53, 149, 173, …. Not sufficient
ii. Prime Numbers q with factors 2 and 3 in q+1 are: 5, 11, 23, 29, 47, 53, 59, ….. Not sufficient

i+ii: 29, 53, … Not sufficient

E
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Re: Is the prime number q equal to 29 ?  [#permalink]

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02 Sep 2019, 12:27
Hi everyone,

Is the prime number q equal to 29 ?

(1) q - 1 has exactly 6 positive factors.
This is valid for both 29 and 53. Hence insufficient

(2) 2 and 3 are prime factors of q + 1
This is both valid for 29 and 23. Hence insufficient

Taken together the statements are valid for 29 and 53.

Hence E
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Re: Is the prime number q equal to 29 ?  [#permalink]

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02 Sep 2019, 19:16
Bunuel wrote:
Is the prime number q equal to 29 ?

(1) q - 1 has exactly 6 positive factors.
(2) 2 and 3 are prime factors of q + 1

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Since we have 1 variable ($$q$$) and 0 equations, D is most likely to be the answer. So, we should consider each condition on its own first.

Condition 1)
We have $$q - 1 = a^5$$ or $$q - 1 =a^2 \cdot b^3$$ for prime numbers $$a$$ and $$b$$.
If $$q - 1 = 2^2 \cdot 7^1$$, then we have $$q = 29$$ and the answer is 'yes'.
If $$q - 1 = 2^2 \cdot 3^1$$, then we have $$q = 13$$ and the answer is 'no'.
Since condition 1) does not yield a unique solution, it is not sufficient.

Condition 2)
Since $$2$$ and $$3$$ are prime factors of $$q+1$$, $$q+1$$ is a multiple of $$6$$.
If $$q+1=30$$, then we have $$q=29$$ and the answer is 'yes'.
If $$q+1=24$$, then we have $$q=23$$ and the answer is 'no'.
Since condition 2) does not yield a unique solution, it is not sufficient.

Conditions 1) & 2)
We have $$q - 1 = a^5$$ or $$q - 1 =a^2 \cdot b^3$$ for prime numbers $$a$$ and $$b$$.
Since $$2$$ and $$3$$ are prime factors of $$q+1$$, $$q+1$$ is a multiple of $$6$$.
If $$q - 1 = 2^2 \cdot 7^1$$, then $$q+1=30$$ is a multiple of $$6$$ and we have $$q=29$$ and the answer is 'yes'.
If $$q - 1 = 2^2 \cdot 13^1$$, then $$q+1=54$$ is a multiple of $$6$$ and we have $$q=53$$ and the answer is 'no'.
Since condition 1) does not yield a unique solution, it is not sufficient.

If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E.
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Re: Is the prime number q equal to 29 ?   [#permalink] 02 Sep 2019, 19:16
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