Question

Choose digits(0to9) for each letter which satisfy the following.Each letter has different value.
9*HATBOX=4*BOXHAT

Answer 1

If we call the number "HAT" x and "BOX" y, the equation becomes

& 9(1000x+y) = 4(1000y+x)

which simplifies to

8996x = 3991y

which on division by 13 becomes

& 692x = 307y

where the coefficients are relatively prime.

This has the obvious solution

& x = 307n, y = 692n for any integer n

The only solution in which x and y are both 3-digit numbers is for
n = 1. Then

& HAT = 307
& BOX = 692

& 9(HATBOX) = 9(307692) = 2769228
& 4(BOXHAT) = 4(692307) = 2769228

The only thing wrong is that A = 0, which violates the conditions.
Since this is the only solution to the equation, there is no solution
to the problem as stated.