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

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Joined: 20 Jul 2017
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Location: India
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Which of the following is one less than square of a prime number?  [#permalink]

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03 Mar 2020, 06:40
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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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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)

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

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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

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

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