# Answer to Question #4501 in Quantitative Methods for Aditi

Question #4501

Choose digits(0to9) for each letter which satisfy the following.Each letter has different value.

9*HATBOX=4*BOXHAT

9*HATBOX=4*BOXHAT

Expert's answer

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.

& 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.

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