BOXHAT * 1000 – HATBOX = BOXHAT000 – HATBOX = BOX000000 – BOX = BOX * 999999
On the other hand, BOXHAT * 1000 – HATBOX = (BOXHAT * 4) * 250 – HATBOX
= (HATBOX * 9) * 250 – HATBOX = 2249 * HATBOX.
This means BOX * 999999 = 2249 * (HAT * 1000 + BOX), i.e. 2249000 * HAT = (999999 – 2249) * BOX = 997750 * BOX.
Find the greatest common divisor of 2249000 and 997750, and divide the both sides by it. Then we get 692 * HAT = 307 * BOX.
So the answer is 9 * 307692 = 4 * 692307.