Question 733668
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            The solution by  @CubeyThePenguin is  INCORRECT.


            I came to bring the correct solution.



<pre>
Let six consecutive integer numbers be  n, n+1, n+2, n+3, n+4, n+5.


The condition says


    n+1 = 11m    (1)       (The number in position two is a multiple of 11)

    n+5 = 10k    (2)        (the number in position six is a multiple of 10)


From equation (2), subtract equation (1), You will get

     4  = 10k - 11m.


Find two numbers that differ in 4 units;  the greater is multiple of 10; the smaller is a multiple of 11.


Thinking 7 seconds (or less), you will guess such numbers: they are  70 and 66.


Therefore, k= 7;  m= 6.


Thus the second number in the sequence is  11*6 = 66;  
     the sixth  number in the sequence is  10*7 = 70.


The consecutive six numbers are


    the indexes :     1     2     3     4     5     6

    the numbers :     65   66    67    68    69    70      <U>ANSWER</U>
</pre>

Solved.


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Such a sequence &nbsp;&nbsp;// (a solution) &nbsp;is &nbsp;NOT &nbsp;UNIQUE.


There are other similar solutions &nbsp;(sequences) &nbsp;with the same property.



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For your safety, &nbsp;ignore the post by &nbsp;@CubeyThePenguin, &nbsp;since his solution is &nbsp;WRONG.