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20 Jan 2023, 12:15
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C
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Difficulty:
45%
(medium)
Question Stats:
62% (02:27) correct
38%
(02:32)
wrong
based on 26
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If a, b, c, d, and n are distinct positive integers greater than 1, is \(a^n – b^n + c^n + d^n\) odd ?
(1) a, b, c and d are squares of consecutive prime numbers
(2) a^4 divided by 200 has the quotient 1
(1) a, b, c and d are squares of consecutive prime numbers
(2) a^4 divided by 200 has the quotient 1
20 Jan 2023, 13:44
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Bunuel
Statement 1
(1) a, b, c and d are squares of consecutive prime numbers
Prime number except for 2 are odd.
As a, b, c and d are squares of consecutive prime numbers, if the series do not start from 2, all the numbers will be odd.
However, if the series starts from 2, one number will be odd and other three numbers will be even.
As there are two possible cases, both of which will give different answers to the question "Is \(a^n – b^n + c^n + d^n\) odd ?" statement 1 by itself is not sufficient. We can eliminate A and D.
Statement 2
(2) a^4 divided by 200 has the quotient 1
This statement tells us that \(a^4\) lies between 200 and 400. However as we don't have any additional information we can eliminate B.
Statement 3
a, b, c and d are squares of prime number, also \(a^4\) lies between 200 and 400.
Let's start with smallest prime number 2
Assume
\(a = 2^2 = 4\)
\(4^4 = 256\)
If a = 3
\(a = 3^2 = 9\)
\(9^4 = 81 * 81\) , this value is definitely greater than 400
Hence only possible value of a = 2
As the series consists one even number and the rest of the terms are odd, we have a definite answer "Is \(a^n – b^n + c^n + d^n\) odd ?" -- Yes !
Option C
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