Strange Number (3)

Assume the number can be written in the form: $\overline{abcdefghij}$ where a is the first digit, b is the second digit, c is the third digit … until j is the tenth digit.

Since the whole number is divisible by 10, that means j = 0.

Since $\overline{abcde}$ is divisible by 5, that means e = 5.

Since $\overline{ab}$ is a multiple of 2, that means b = (2, 4, 6 or 8)

Since $\overline{abcd}$ is a multiple of 4, that means d = (2, 4, 6 or 8)

Since $\overline{abcdef}$ is a multiple of 6, that means f = (2, 4, 6 or 8)

Since $\overline{abcdefgh}$ is a multiple of 8, that means d = (2, 4, 6 or 8)

That means a, c, g and i are odd numbers. Therefore:

a = (1, 3, 7 or 9)

c = (1, 3, 7 or 9)

g = (1, 3, 7 or 9)

i = (1, 3, 7 or 9).

In order to have $\overline{abcd}$ a multiple of 4, therefore $\overline{cd}$ must be a multiple of 4. Since c is odd, therefore d can only be (2 or 6).

In order to have $\overline{abcdefgh}$ a multiple of 8, therefore $\overline{fgh}$ must be a multiple of 8. Since f is even, g is odd, therefore d can only be (2 or 6).

Since d and h are 2 and 6 in a certain order that means b and f can only be 4 and 8 in some order.

Let’s summarize the possibilities for each digit:

a = (1, 3, 7 or 9)

b = (4 or 8)

c = (1, 3, 7 or 9)

d = (2 or 6)

e = 5

f = (4 or 8)

g = (1, 3, 7 or 9)

h = (2 or 6)

i = (1, 3, 7 or 9).

j = 0

Let’s start by building the numbers by all possibilities and exclude the wrong ones :

The first digit a can only be 1, 3, 7 or 9.

The first 2 digits ab can only be: 14, 18, 34, 38, 74, 78, 94 or 98.

The first 3 digits $\overline{abc}$ is a multiple of 3. Therefore, a + b + c should be a multiple of 3. Also we should exclude the numbers where a = c.

The possibilities for $\overline{abc}$ are: 147, 183, 189, 381, 387, 741, 783, 981 or 987.

Let us consider now the next 3 digits $\overline{def}$ . Since $\overline{abcdef}$ is a multiple of 6, therefore $\overline{abcdef}$ should be a multiple of both 2 and 3. Since f can only be 4 or 8 therefore it’s already a multiple of 2 and we can ignore this case. We now have $\overline{abcdef}$ a multiple of 3. Since $\overline{abc}$ is a multiple of 3, therefore we should have def a multiple of 3. That means d + e + f should be a multiple of 3.

The possibilities for def are now: 258 or 654.

If we put them together and exclude the case where digits are repeated we get for abcdef.

The possibilities for $\overline{abcdef}$ are: 147258, 183654, 189654, 381654, 387654, 741258, 783654, 981654 or 987654.

To find $\overline{abcdefg}$ which are multiple of 7 we should calculate ( $\overline{abcdef}$ X 10) mod 7.

If result is 0, we should append 7 if it was not used in $\overline{abcdef}$ .

If result is 4, we should append 3 if it was not used in $\overline{abcdef}$ .

If result is 5, we should append 9 if it was not used in $\overline{abcdef}$ .

If result is 6, we should append 1 if it was not used in $\overline{abcdef}$ .

For all other results, we should reject $\overline{abcdef}$ .

The possibilities for $\overline{abcdefg}$ are: 1472583, 1836541, 3816547 or 7836549.

To have the eight digits number $\overline{abcdefgh}$ a multiple of 8, we should have only $\overline{fgh}$ as a multiple of 8.

Since f is even, we can then only check if $\overline{gh}$ is a multiple of 8.

Since h can only be 2 or 6 we have to check which one was not previously used, append it and check the last two digits if they are a multiple of 8. Doing this for all the previous numbers we got so far for $\overline{abcdefg}$ we get:

The only possibility for $\overline{abcdefgh}$ are: 38165472.

The last step is to find the missing non-zero digit to append and then append a zero at the end. This will give us: 3816547290.

$Answer = \boxed{3816547290}$