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31 Jan 2023, 12:15
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B
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
75%
(hard)
Question Stats:
49% (02:21) correct
51%
(02:28)
wrong
based on 65
sessions
History
Date
Time
Result
Not Attempted Yet
If y is a positive integer, is (y^4+7)^2 divisible by 4?
(1) √y has exactly three Prime Factors
(2) All the prime Factors of y^7 are greater than 3
(1) √y has exactly three Prime Factors
(2) All the prime Factors of y^7 are greater than 3
11 Feb 2023, 11:21
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Bunuel
If a number is divisible by 4 then it will be an even number.
(1) √y has exactly three Prime Factors
\(\sqrt y\) is square of a prime, as only square of primes have total 3 factors.
If \(\sqrt y =4 \) then \(y = 16 \)
In this case \((y^4+7)^2\) turns out be odd and is NOT divisble by \(4 \)
If \(\sqrt y =9 \) then \(y = 81 \)
\((81^4+7)^2 = (3^{16}+7)^2\) . Little calculation shows that this is divisble by \(4 \)
INSUFF.
(2) All the prime Factors of \(y^7 \) are greater than \(3\)
All prime numbers greater than \(3\) are odd,So \(y\) will be odd and the expression \((y^4+7)^2 \) will be of the form:
\((3^4+7)^2 =\hspace{2mm}\)Divisible by \(4 \)
\((5^4+7)^2 =\hspace{2mm}\)Divisible by \(4\)
\((7^4 +7)^2=\hspace{2mm}\)Divisible by \(4\)
\((11^4 +7)^2=\)Divisible by \(4\)
etc.
Note: One needs to only calculate the last two digits of the above expression to determine whether it's divisble by \(4.\)
Hence we can definitely answer yes to the question
SUFF.
Ans B
Hope it's clear
12 Feb 2023, 12:05
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A very good question to test odd and even concept
Odd+Odd = even number
Even number is divisible by 4
7 is odd number, therefore the variable y has to be also odd. To verify we can go through the options to check whether the variable y is odd or even.
Explanation
Option A
(1) √y has exactly three Prime Factors
does not tell anything about the Variable y except it’s roomy having 3 prime factors , which can be VARIABLe PERFECT SQUARES .. starting from 4, 9,etc . Variable y can be odd or even.
Hence insufficient
Option B
(2) All the prime Factors of y^7 are greater than 3
2 is the only even prime number, which is now excluded from the list . Therefore any perfect square with odd number can take the value of y
Hence sufficient
Answer option B
Odd+Odd = even number
Even number is divisible by 4
7 is odd number, therefore the variable y has to be also odd. To verify we can go through the options to check whether the variable y is odd or even.
Explanation
Option A
(1) √y has exactly three Prime Factors
does not tell anything about the Variable y except it’s roomy having 3 prime factors , which can be VARIABLe PERFECT SQUARES .. starting from 4, 9,etc . Variable y can be odd or even.
Hence insufficient
Option B
(2) All the prime Factors of y^7 are greater than 3
2 is the only even prime number, which is now excluded from the list . Therefore any perfect square with odd number can take the value of y
Hence sufficient
Answer option B
