Find the letters 9 x HATBOX = 4 x BOXHAT

Each letter must represent a digit from 1 to 9 with no two letters representing the same digit.

Solution

We have: 9 x HATBOX = 4 x BOXHAT

set "HAT" =x and "BOX"=y(x,y has 3-digit)

the equation becomes

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

8996x = 3991y

which on division by 13 becomes

692x = 307y

where the coefficients are relatively prime.

This has the obvious solution we can set as:

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

Thus: H=3,A=0,T=7,B=6,O=9,X=2