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If n is a prime number, is n − 1 equal to the square of an integer?

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Bunuel
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If n is a prime number, is n − 1 equal to the square of an integer? [#permalink] New post   29 Jun 2020, 01:43
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Competition Mode Question



If n is a prime number, is n − 1 equal to the square of an integer?

(1) n < 10
(2) n is a solution of x^2 − 2x − 35 = 0.
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B
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Re: If n is a prime number, is n − 1 equal to the square of an integer? [#permalink] New post   29 Jun 2020, 02:17
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Explanation:
Given: n is a prime number
Find: (n-1) is square of an integer

Statement 1: n<10
if n=5 , yes (n-1) is square of an integer
if n=2,3,7 No.
Not Sufficient.

Statement 2: x^2 − 2x − 35 = 0
(x-7)(x+5)=0
x=7,-5 ; n=7,-5
(n-1) = 7-1 = 6 ; No
(-5-1) = -6 ; No

Sufficient

IMO-B
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Re: If n is a prime number, is n − 1 equal to the square of an integer? [#permalink] New post   29 Jun 2020, 02:18
If n is a prime number, is n − 1 equal to the square of an integer?

Solution : Key Highlight: Prime numbers are always positive

(1) n < 10
Prime number less than 10 = 2,3,5&7
where, 2-1 = 1 is a square of integer 1.
where, 3-1 = 2 is not a square of any integer.
where, 5-1 = 4 is a square of integer 2.
where, 7-1 = 6 is not a square of any integer.
(Not Sufficient)

(2) n is a solution of x^2 − 2x − 35 = 0.
x^2 − 7x + 5x − 35 = 0.
x(x - 7) + 5 (x − 7) = 0.
( X - 7 ) (X + 5)
x = 7 & -5
since x is a prime number
x is only 7
but X-1 = 6 which is NOT a square of any integer.
(Sufficient)

IMO B
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Re: If n is a prime number, is n − 1 equal to the square of an integer? [#permalink] New post   29 Jun 2020, 06:01
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The question has already put a constraint on the value of n.
n is a prime number, which means \(n > 0\)

Option A:
Values possible for n = 2,3,5,7
Values of n - 1 = 1,2,4,6

Now, 1 and 4 are square of an integer.

INSUFFICIENT

Option B:
\(x^2 - 2x - 35 = 0\)
\((x-7)(x+5) = 0\)
\(x = 7\) OR \(x = -5\)

N is a solution of this equation, and we know that N>0
Only possible value of N = 7
N - 1 = 6
6 is not the square of a positive integer.

SUFFICIENT

OA, B
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Re: If n is a prime number, is n − 1 equal to the square of an integer? [#permalink] New post   29 Jun 2020, 09:45
IMO-B

Statement 1:-
When n<10
prime no will be 2,3,5,7
2-1 is not square of any integer but 5-1 is square of integer 2
Insufficient

Statements-2
X^2-2x-35=0
(X-7)(x+5)=0
Value of x=-5&7
By the usual definition of prime for integers, negative integers can not be prime. By this definition, primes are integers greater than one with no positive divisors besides one and itself. Negative numbers are excluded.

So value of n = 7

Sufficient

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Re: If n is a prime number, is n − 1 equal to the square of an integer? [#permalink] New post   29 Jun 2020, 10:06
TIME 1:49
IMO B

Kindly see the attachment. I have documented the steps that I took during exam mode for this question.
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Re: If n is a prime number, is n − 1 equal to the square of an integer? [#permalink] New post   29 Jun 2020, 12:02
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If n is a prime number, is n − 1 equal to the square of an integer?

(1) n < 10
(2) n is a solution of x^2 − 2x − 35 = 0.

1) prime numbers less than 10 are 2, 3, 5, 7. If n = 5, yes. For other 3, the answer is no. Not sufficient

2) Solving the equation we get, x = 7, or x = -5. Since n is prime, so it cannot take the negative value. 7 is the only option. sufficient.

B is the answer.
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Re: If n is a prime number, is n − 1 equal to the square of an integer? [#permalink] New post   29 Jun 2020, 18:13
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Given: n is prime.
Asked: Is n-1 is square of an integer?

One more understanding: n is prime means it is integer.

St. 1
n<10

Take n =5, n-1= 4 which is square of 2. Therefore, yes.
Take n= 9, n-1= 8 which is not an square of an integer. Therefore, no.

A and D is out.

St. 2
Solution of given quadratic equation is 7 and -5
but n is prime so it is positive integer also.
therefore, n= 7
but n-1= 6 is not a square of an integer.

Answer is B.
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Re: If n is a prime number, is n − 1 equal to the square of an integer? [#permalink] New post   29 Jun 2020, 19:43
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Quote:
If n is a prime number, is n − 1 equal to the square of an integer?

(1) n < 10
(2) n is a solution of x^2 − 2x − 35 = 0.


n=p; p-1=integer^2?

(1) insufic

0<n=p<10: 2,3,5,7
n-1: 1,2,4,6

(2) sufic

x^2 − 2x − 35 = 0, (x-7)(x+5)=0,
x>0: x=7=n=p, n-1=6 not integer^2

Ans (B)
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Re: If n is a prime number, is n − 1 equal to the square of an integer? [#permalink] New post   29 Jun 2020, 20:33
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If n is a prime number, is n − 1 equal to the square of an integer?

(1) n < 10
(2) n is a solution of x^2 − 2x − 35 = 0.

1) n can be 2,3,5,7
when n=3, n-1 is not square of an integer
when n=5, n-1 is square of an integer
insufficient

2) x^2-2x_35=0
n=7,-5
when n=7, n-1 is not square of an integer
sufficient

Ans B
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Re: If n is a prime number, is n − 1 equal to the square of an integer? [#permalink] New post   01 Aug 2020, 10:29
Bunuel wrote:

Competition Mode Question



If n is a prime number, is n − 1 equal to the square of an integer?

(1) n < 10
(2) n is a solution of x^2 − 2x − 35 = 0.


I had a doubt in statement 1. If n-1 is square if an interger. When n<10, only n=5. We can correctly find the answer then why is statement 1 not sufficient?

Also for statement 2 we have two values, how can we ne certain of one single value that satisfies?

Thank you in advance for answering my doubt.

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Re: If n is a prime number, is n − 1 equal to the square of an integer? [#permalink]
01 Aug 2020, 10:29
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Bunuel
Math Expert
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