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If a and b are positive integers, is a^4b^4 divisible by 4?
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Updated on: 07 Oct 2018, 06:04
Question Stats:
52% (01:57) correct 48% (01:01) wrong based on 32 sessions
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Originally posted by gmatbusters on 06 Oct 2018, 10:25.
Last edited by gmatbusters on 07 Oct 2018, 06:04, edited 2 times in total.
Renamed the topic and edited the question.



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Re: If a and b are positive integers, is a^4b^4 divisible by 4?
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06 Oct 2018, 10:25
Official Explanation: \(a^4b^4=(ab)(a+b)(a^2+b^2)\) Statement 1 : a+b is divisible by 4, a+b = 4k Since, \(a^4b^4=(ab)(a+b)(a^2+b^2)\), we get, \(a^4b^4=4k(ab)(a^2+b^2)\) Hence it is divisible by 4, SUFFICIENT Statement 2 : \(a^2+b^2\) , when divided by 4 gives 2 as remainder, Hence \(a^2+b^2\) is Even. It means either both a and b are EVEN or ODD. When both are Even, a = 2m, b = 2n: \(a^4b^4 = 4(m^4n^4)\), divisible by 4 When both are oddm a = 2m +1, b = 2n+1: \(a^4b^4=(ab)(a+b)(a^2+b^2)\) =\(4(mn)(m+n+1)(a^2+b^2)\), hence divisible by 4 SUFFICIENTAnswer D
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Re: If a and b are positive integers, is a^4b^4 divisible by 4?
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06 Oct 2018, 10:29
a4−b4 = (a+b)*(ab) (a2+b2)
Option A: a+b is divisible by 4 since a and b are integers, it definitely answers that a4b4 will be divisible by 4
2) The remainder is 2 when a2+b2 is divisible by 4 which means a2+b2=. 4k +2
so it becomes (a+b)(ab) (4k+2) can't say that it is divisible by 4
A is the answer



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Re: If a and b are positive integers, is a^4b^4 divisible by 4?
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06 Oct 2018, 10:31
A
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Re: If a and b are positive integers, is a^4b^4 divisible by 4?
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06 Oct 2018, 10:35
a^4  b^4 = (ab)(a+b)(a^2b^2)
St 1 says a+4 div by 4  sufficient
St 2 says a^2+b^2mod 4 = 2 (remainder 2) which means a and b are either both odd, or both even Both cases imply a+b is even Hence product of two even numbers is div by 4
D



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Re: If a and b are positive integers, is a^4b^4 divisible by 4?
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06 Oct 2018, 10:36
a^4b^4 can be factorized as (a^2b^2)*(a^2+b^2)... (ab)*(a+b)*(a^2+b^2)
St.1 says a+b is divisible by 4. Hence sufficient.
St.2 says remainder is 2 when a^2+b^2 is divided by 4.. hence it is divisible by 2. It could have both even or both odd parts adding up to an even total... Hence either ways a+b and ab will be even. Therefore the whole will be divisible by 4.
St2. Is also sufficient.
Hence option (d) is correct.
Best, G
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Re: If a and b are positive integers, is a^4b^4 divisible by 4?
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06 Oct 2018, 10:38
answer is D: a and b statement 1: (a^4b^4) = (a^2+b^2) (a+b)(ab) if a+ b is divisible by 4 then the statement is divisible by 4
statment 2:plug in numbers. 3^2+3^2 = 18 (which has a remainder of 2 when divided by 4) and 8181 = 0 which is divisible by 4 5^2 + 1^2 = 26 and 5251 is divisible by 4



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Re: If a and b are positive integers, is a^4b^4 divisible by 4?
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06 Oct 2018, 10:53
Answer D
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Re: If a and b are positive integers, is a^4b^4 divisible by 4?
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06 Oct 2018, 10:59
a^4b^4=(a^2+b^2)(a+b)(ab) 1) suff 2)Suff (a^2+b^2) div 4..... rem 2 .....===> (a^2+b^2) is divisible by 2 not by 4 case 1 : so a, b both even .... then (a+b) & (ab) both div by 2 case 2 : so a, b both odd .... then (a+b) & (ab) both div by 2 Hence ans D
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