Prove that ` sum_(k=0)^(9) x^(k)"divides" sum_(k=0)^(9) x^(kkkk)`

Explanation

Let `S_(1) = sum_(k=0)^(9) x^(kkkk)= x^(0) + x^(1111) + x^(2222) + ...+ x^(9999)`
and ` S_(2) = sum_(k=0)^(9) x^(k)= x^(0) + x^(1) + x^(2) + ...+ x^(9)`
Now , ` S_(1) - S_(2) = sum_(k=0)^(9) (x^(kkkk) - x^(k)) = sum_(k=0)^(9) x^(k) (x^(10))^(kkkk) -1`
`= [(x^(10))^(kkk) -1] sum_(k=0)^(9) x^(k) = lambda sum_(k=0) ^(9) x^(k)`
` rArr S_(1) -S_(2) = lambda S_(2) rArr S_(1) (1+ lambda) S_(2)` .
Hence , ` sum_(k=0)^(9) x^(kkkk)` is divisible by ` sum_(k=0)^(9) x^(k)`.