Intermediate Combinatorics

There are 9 digits to share between the two numbers. We want the two numbers to be as similar in size as possible so their number of digits needs to be as close as possible. 9 is odd so they can't have the same number of digits but one can have 5 and the other one 4.

The 5 digit number will always be bigger than the 4 digit number but to minimise the difference we make the 4 digit number as large as possible and the 5 digit number as small as possible. Therefore the numbers are 9876 and 12345 and the difference is 2469.

2469= $(1 \times 7^4)+(0 \times 7^3)$

$+(1 \times 7^2)+(2 \times 7^1)+(5 \times 7^0)$

So 2469 in base 7 is 10125.

1+0+1+2+5=9 so the answer is $\boxed{9}$ .

Step 1 : For least difference :

LET the 2 numbers be A & B. ( where A > B ).

For A, it should be the least possible 5-digit number.

For B, it should be the max. possible 4-digit number.

So, A = 12345 , B = 9876. We get A - B = 2469.

Step 2 : For Conversion :

7 ^ 0 = 1 .

7 ^ 1 = 7 .

7 ^ 2 = 49 .

7 ^ 3 = 343 .

7 ^ 4 = 2401 .

2469 = 1 * ( 7 ^ 4 ) + 0 * ( 7 ^ 3 ) + 1 * ( 7 ^ 2 ) + 2 * ( 7 ^ 1 ) + 5 * ( 7 ^ 0 )

( 2469 )-base 10 = ( 10125 )-base 7.

Final step : Sum of digits = 1 + 0 + 1 + 2 + 5 = 9.