Answers

2015-05-05T14:04:18+05:30
Recall from little fermat we have p\mid(a^{p}-a) for all integers a and prime p

\frac{k^7}{7}+\frac{k^5}{5}+\frac{2k^3}{3}-\frac{k}{105}=\frac{15k^7+21k^5+70k^3-k}{105}

105=3\cdot5\cdot7, so it is sufficient to show that 15k^7+21k^5+70k^3-k is divisible by 3,5,7

15k^7+21k^5+70k^3-k\equiv 0+0+k^3-k\equiv k^3-k\equiv{0}\pmod{3} 
15k^7+21k^5+70k^3-k\equiv 0+k^5+0-k\equiv 0\equiv k^5-k\equiv 0\pmod{5} 
15k^7+21k^5+70k^3-k\equiv k^7+0+0-k\equiv 0\equiv k^7-k\equiv 0\pmod{7} 

Therefore 15k^7+21k^5+70k^3-k is divisible by 3,5,7 and consequently the expression main question is an integer for all value sof k
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