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16 Sep 2014, 00:23
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Is \(\frac{7^7}{7^x}\) an integer?
(1) \(0 \le x \le 7\)
(2) \(|x| = x^2\)
(1) \(0 \le x \le 7\)
(2) \(|x| = x^2\)
16 Sep 2014, 00:23
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Official Solution:
Is \(\frac{7^7}{7^x}\) an integer?
The expression \(\frac{7^7}{7^x} = 7^{7-x}\) will result in an integer if \(x\) is an integer and \(x\) is less than or equal to 7. For example, if \(x\) takes values of 7, 6, 5, ..., 0, -1, ... and so on, then \(7^{7-x}\) will always be an integer. However, for values of \(x\) such as 8, 9, 10, ... or any non-integer, such as 3.5, or -2.5, \(7^{7-x}\) will not result in an integer.
(1) \(0 \le x \le 7\).
If \(x\) is an integer from this range, \(\frac{7^7}{7^x}\) will be an integer but if \(x\) is a non-integer from this range, \(\frac{7^7}{7^x}\) will NOT be an integer.
(2) \(|x| = x^2\)
Square \(|x| = x^2\);
\(x^2 = x^4\);
\(x^2(x^2 - 1) = 0\)
\(x^2(x - 1)(x + 1) = 0\);
\(x = 0\), 1, or -1. For each of those values \(\frac{7^7}{7^x}\) IS an integer. So, we have an YES answer.
Answer: B
Is \(\frac{7^7}{7^x}\) an integer?
The expression \(\frac{7^7}{7^x} = 7^{7-x}\) will result in an integer if \(x\) is an integer and \(x\) is less than or equal to 7. For example, if \(x\) takes values of 7, 6, 5, ..., 0, -1, ... and so on, then \(7^{7-x}\) will always be an integer. However, for values of \(x\) such as 8, 9, 10, ... or any non-integer, such as 3.5, or -2.5, \(7^{7-x}\) will not result in an integer.
(1) \(0 \le x \le 7\).
If \(x\) is an integer from this range, \(\frac{7^7}{7^x}\) will be an integer but if \(x\) is a non-integer from this range, \(\frac{7^7}{7^x}\) will NOT be an integer.
(2) \(|x| = x^2\)
Square \(|x| = x^2\);
\(x^2 = x^4\);
\(x^2(x^2 - 1) = 0\)
\(x^2(x - 1)(x + 1) = 0\);
\(x = 0\), 1, or -1. For each of those values \(\frac{7^7}{7^x}\) IS an integer. So, we have an YES answer.
Answer: B
28 Nov 2014, 07:31
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need to be careful..........
x is not integer, x can be 1.2, 3.5 5.6 anything. Any decimal in (0,7).....
CAREFUL!!!!!!!!!!!!!!!!!!!!!!!!!!
hence using 1st option it can be fraction!!!! Grrr.....
x is not integer, x can be 1.2, 3.5 5.6 anything. Any decimal in (0,7).....
CAREFUL!!!!!!!!!!!!!!!!!!!!!!!!!!
hence using 1st option it can be fraction!!!! Grrr.....
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