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09 Jun 2022, 00:12
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C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
65%
(hard)
Question Stats:
61% (02:19) correct
39%
(02:30)
wrong
based on 170
sessions
History
Date
Time
Result
Not Attempted Yet
\(x=6^8-8^4\). Which of the following is NOT a divisor of \(x\)?
a) \(2\)
b) \(10\)
c) \(33\)
d) \(35\)
e) \(77\)
a) \(2\)
b) \(10\)
c) \(33\)
d) \(35\)
e) \(77\)
Updated on: 24 Oct 2023, 15:15
Originally posted by EthanTheTutor on 09 Jun 2022, 00:18.
Last edited by EthanTheTutor on 24 Oct 2023, 15:15, edited 7 times in total.
Last edited by EthanTheTutor on 24 Oct 2023, 15:15, edited 7 times in total.
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EthanTheTutor
Method 1 - Algebra:
\(x=6^8-8^4\)
\(x=2^8\cdot3^8-2^{12}\)
\(x=2^8(3^8-2^4)\)
\(x=2^8(3^4-2^2)(3^4+2^2)\)
\(x=2^8(3^2-2)(3^2+2)(3^4+2^2)\)
\(x=2^8\cdot7\cdot11\cdot85\)
\(x=2^8\cdot7\cdot11\cdot5\cdot17\)
\(33=3\cdot11\) cannot be formed using the prime factors of \(x\). Answer C.
(a shortcut here is to realize that 6 is a multiple of 3, and 8 is not a multiple of 3, so we're adding a multiple of 3 to a non-multiple of 3; the result can never be a multiple of 3)
Method 2 - Reasoning from the answer choices:
a) \(2\)
b) \(10=2\cdot5\)
c) \(33=3\cdot11\)
d) \(35=5\cdot7\)
e) \(77=7\cdot11\)
If \(x\) were not divisible by \(2\), then both answers A and B would be correct. This is impossible, thus \(x\) must be divisible by \(2\).
If \(x\) were not divisible by \(5\), then both answers B and D would be correct. This is impossible, thus \(x\) must be divisible by \(5\).
If \(x\) were not divisible by \(7\), then both answers D and E would be correct. This is impossible, thus \(x\) must be divisible by \(7\).
If \(x\) were not divisible by \(11\), then both answers C and E would be correct. This is impossible, thus \(x\) must be divisible by \(11\).
The only prime factor in the answer choices that does not produce a contradiction is \(3\). Thus, \(x\) is not divisible by \(3\). Answer C.
15 Dec 2023, 05:42
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2^8(3^8-2^4)
2^8(6561-16)
2^8 * 6545
check options
2 divisor
2*5 divisor
11*3, cannot be divided by 3
7*5 divisor
11*7 divisor
2^8(6561-16)
2^8 * 6545
check options
2 divisor
2*5 divisor
11*3, cannot be divided by 3
7*5 divisor
11*7 divisor
