SOLUTION: find the smallest 7 digit mumber formed by 0,1,2,3 and 5 completely divisible by 198?

Question 874953: find the smallest 7 digit mumber formed by 0,1,2,3 and 5 completely divisible by 198?

Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
The way I understand the problem,
all the digits listed are required (they have to be in the number),
and no other digits are allowed.

so the number must be a multiple of , and .

Including only allowed digits, to be a multiple of ,
the number must end in or .

To be a multiple of the number's digits must add to a multiple of .
Since the required digits add up to ,
the sum of all digits cannot add to .
With two more digits the only other multiple of possible for that sum is
.
The two extra digits must add to , so they must be
and .
So, the seven digits are
, , , , , , and .

For a 7 digit number to be a multiple of ,
the sum of the 1st, 3rd, 5th, and 7th digits, ,
and the sum of the 2nd, 4th, and 6th digits, ,
must differ by or a multiple of .
In this case, since both sums add to ,
the difference could only be or ,
not or other multiples.
So, while we know that ,
the sums must also satisfy
or or .
Since the systems formed by or ,
along with , do not have integer results,
the system we have to solve is
.
Its solution is .
We need to separate the digits to be used,
, , , , , , and ,
into two groups:
a group of 4 digits that add up to ,
and a group of 3 digits that add up to .
With those digits, the only ways to get a group of 3 digits that add up to is
or .
One of those groups of digits will include , , and ,
and the other group will include , , and .
The required will have to be added to one of those groups to get the group of 4 digits that will be the 1st, 3rd, 5th, and 7th digits.

To get the smallest possible number, we would want the first digit to be ,
so the 1st, 3rd, 5th, and 7th digits must include
, , , and .
We are choosing as the 1st digit.
We need to use as the 7th and last digit,
because the number must be a multiple of .
Regarding the 3rd, and 5th digits, the number will be smallest if
is the 3rd digit,
and is the 5th digit.
So the 1st, 3rd, 5th, and 7th digits are
, , , and , in that order.

Regarding the 2nd, 4th, and 6th digits, which must include
, , and ,
we get the smallest number if we use them in that order.

So, the number is .
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