MHB Find the Smallest of A and B: $A-B=98$ with Multiple of 19 Digit-sum

[Albert1]

$(1)A,B \in N,A-B=98$

(2)All the digit-sum of $A$ and $B$ are multiple of 19

please find the smallest of $A\,\, and \,\, B$

 

[kaliprasad]

$(1)A,B \in N,A-B=98$

(2)All the digit-sum of $A$ and $B$ are multiple of 19

please find the smallest of $A\,\, and \,\, B$

Spoiler

we have B mod 9 = x if sum of digits of a = 19 x
so (B + 98) mod 9 = x+ 8
for A and B both divisible by 19 we should have
so A Mod 9 is one less than B mod 9 with the proviso that b mod 9 = 0 => a mod 9 = 8

so smallest possible candidate should be digit sum of B= 38 and A = 19 as digit sum of B mod 9 is 1 more than digit sum of A mod 9
Smallest possible B = 29999 giving A = 30097 and it meets criteria
so B = 29999, A = 30097 is smallest solution,