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Which of the following is one less than square of a prime number?

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Which of the following is one less than square of a prime number?  [#permalink]

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New post 03 Mar 2020, 06:40
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A
B
C
D
E

Difficulty:

  55% (hard)

Question Stats:

45% (01:57) correct 55% (02:07) wrong based on 31 sessions

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Which of the following is one less than square of a prime number?

A. 1295
B. 2303
C. 5220
D. 6560
E. 7920
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Re: Which of the following is one less than square of a prime number?  [#permalink]

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New post 03 Mar 2020, 09:59
Dillesh4096 wrote:
Which of the following is one less than square of a prime number?

A. 1295
B. 2303
C. 5220
D. 6560
E. 7920


Option + 1 = Prime^2

Now, Check the options -

(A) \(\sqrt{1296 } = 36\)
(B) \(\sqrt{2304} = 48\)
(C) \(\sqrt{5221} = 72.xx\)
(D) \(\sqrt{6561} = 81\)
(E) \(\sqrt{7921} = 89\) (Prime Number)

Thus, Answer must be (E)
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Re: Which of the following is one less than square of a prime number?  [#permalink]

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New post 04 Mar 2020, 08:27
1
N = p^2-1
= (6n+/-1)^2 -1
= 6n(n+2)

so the N must be divisible by 6, hence 3
adding the digits and checking divisibility by 3, only C & E are divisible by 3.

for C , N = 5221 72..something
for E, N = 7921 = 89^2

Hence E is the right answer

Dillesh4096 wrote:
Which of the following is one less than square of a prime number?

A. 1295
B. 2303
C. 5220
D. 6560
E. 7920

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Joined: 20 Jul 2017
Posts: 1506
Location: India
Concentration: Entrepreneurship, Marketing
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Re: Which of the following is one less than square of a prime number?  [#permalink]

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New post 06 Mar 2020, 04:40
Dillesh4096 wrote:
Which of the following is one less than square of a prime number?

A. 1295
B. 2303
C. 5220
D. 6560
E. 7920


Any prime number more than 3 can be written in the form \(6n ± 1\)

Let \(x\) = \(6n ± 1\)
Square of a prime number = \(x^2 = (6n ± 1)^2 = 36n^2 ± 12n + 1\)

One less than the square of prime number = \(x^2 - 1 = 36n^2 ± 12n = 12n(3n ± 1)\)

Note that \(12n(3n ± 1)\) is always divisible by \(24\).
Case 1: If \(n\) is odd, \(12n(3n ± 1)\) = \(12\)*odd*even --> always divisible \(24\)
Case 2: If \(n\) is even, \(12n(3n ± 1)\) = \(12\)*even*odd --> always divisible \(24\)


--> All the options are divisible by \(24\)
--> They must be divisible by factors of 24 also = {\(1, 2, 3, 4, 6, 8, . . . \)}

Check for 3 --> {C, E} are divisible & {A, B, D} are NOT divisible. So, Eliminate {A, B, D}
Both C & E are divisible by 4 [Last 2 digits should be divisible by 4: 20/4 = 5]

--> Check for 8 [Last 3 digits should be divisible by 8
C. 220/8 --> NOT divisible
E. 920/8 --> Divisible

Option E
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Re: Which of the following is one less than square of a prime number?   [#permalink] 06 Mar 2020, 04:40

Which of the following is one less than square of a prime number?

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