# What is the small

We have the digits 2, 3, 4, 5, 6. Using these digits, we need to form the smallest 5-digit number divisible by 11.

If we had had the (smallest) number as abcde, this would have satisfied the criteria:

(a + c + e) = (b + d) (mod 11)

Adding these digits, 2 + 3 + 4 + 5 + 6 = 20, so this means that (a + c + e) and (b + d) must be either equal or the difference must come out to be 11 or a multiple of 11. Since, the numbers are small, we can say that the difference won’t come out to be greater multiples of 11.

Observe that from the given digits, (a + c + e) = (b + d) = 10

With the given digits, b + d = 10 is possible.

So, if (b + d) = 10, then (a + c + e) = 10

Such that, (a + c + e) = (b + d) (mod 11)

10 = 10 (mod 11)

From the given digits, {b, d} = {4, 6}

{a, c, e} = {2, 3, 5}

The number is abcde, that is, 24365, which is the smallest possible 5-digit number containing each of the digit 2, 3, 4, 5, 6.

Thus, the number is 24365.

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