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09 Sep 2019, 22:51
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D
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
95%
(hard)
Question Stats:
38% (02:11) correct
62%
(02:26)
wrong
based on 202
sessions
History
Date
Time
Result
Not Attempted Yet
The number of integers x such that \(0 ≤ 2^x ≤ 200\), and \(2^x+2\) is perfectly divisible by either 3 or 4, is
A. 2
B. 3
C. 4
D. 5
E. 6
Posted from my mobile device
A. 2
B. 3
C. 4
D. 5
E. 6
Posted from my mobile device
09 Sep 2019, 23:19
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Dillesh4096
There are two conditions
(A)\(0 ≤ 2^x ≤ 200\), here x can take values 0 to 7 as \(2^0=1 ..2^7=128\) and
(B) \(2^x+2\) .....so, the values are \(2^0+2...2^1+2..2^7+2\)
Divisible by 3-- X as even number will make \(2^x+2\) divisible by 2 due to the remainder property ( as \(2^{(even)}\) will leave a remainder of 1 when divided by 3 and when you add 2 to it, the term is divisible by 3) or you can check with pattern too.. \(2^1+2=4\)..No, \(2^2+2=6\)..Yes
So, x as 0, 2, 4, 6 are possible.
Divisible by 4-- All terms having x greater than 1 will make 2^x divisible by 4, so when you add 2 it, it will never be divisible by 4..
.for example \(2^3+2..... 2^3\) is divisible by 4, so \(2^3+2\) will NOT be divisible..
check for just 0 and 1...\(2^0+2=3.\)..NO AND \(2^1+2=4\)..YES
So x as 1 is possible
Total values - 0, 1, 2, 4, 6. Hence 5 is the answer
D
Updated on: 20 Sep 2019, 05:30
Originally posted by EncounterGMAT on 09 Sep 2019, 23:27.
Last edited by EncounterGMAT on 20 Sep 2019, 05:30, edited 1 time in total.
Last edited by EncounterGMAT on 20 Sep 2019, 05:30, edited 1 time in total.
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Dillesh4096
\(0 ≤ 2^x ≤ 200\) => x must be 0,1, 2, 3, 4, 5, 6 and 7
If x=0, \(2^0+2\) is divisible by 3.........................(1)
If x=1, \(2^1+2\) is divisible by 4.........................(2)
If x=2, \(2^2+2\) is divisible by 3.........................(3)
If x=3, \(2^3+2\) not divisible by 3 or 4
If x=4, \(2^4+2\) is divisible by 3.........................(4)
If x=5, \(2^5+2\) not divisible by 3 or 4
If x=6, \(2^6+2\) is divisible by 3.........................(5)
Therefore, there are 5 values of x satisfying the conditions stated in the question. Thus, the answer is option D
