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The solution by @CubeyThePenguin is INCORRECT.
I came to bring the correct solution.
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 ANSWER
Solved.
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Such a sequence // (a solution) is NOT UNIQUE.
There are other similar solutions (sequences) with the same property.
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For your safety, ignore the post by @CubeyThePenguin, since his solution is WRONG.