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A four digit number,
n is written as
WXYZ where
W,X,Y,Z are all different odd digits. Furthermore
n is divisible by each of
W,X,Y,Z. What is the largest possible value for
W-X+Y-Z ?

A four digit number, $n$ is written as $W X Y Z$ where $W, X, Y, Z$ are all different odd digits. Furthermore $n$ is divisible by each of $W, X, Y, Z$ . What is the largest possible value for $W-X+Y-Z$ ?

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Transformations of linear functions

Full solution

Q. A four digit number,

n

is written as

W X Y Z

where

W, X, Y, Z

are all different odd digits. Furthermore

n

is divisible by each of

W, X, Y, Z

. What is the largest possible value for

W-X+Y-Z

?

Identify Largest Odd Digit for $W$ : Identify the largest possible odd digit for $W$ to maximize the value of $W-X+Y-Z$ , considering that $WXYZ$ must be divisible by $W$ . $\newline$ Since $W$ is the most significant digit of a four-digit number, it cannot be $0$ , and since it must be odd, the largest possible value for $W$ is $9$ .
Determine Largest Odd Digit for $\newline$ $Y$ : Determine the largest possible odd digit for $\newline$ $Y$ to maximize the value of $\newline$ $W-X+Y-Z$ , considering that $\newline$ $WXYZ$ must be divisible by $\newline$ $Y$ . $\newline$ Since $\newline$ $Y$ must be odd and different from $\newline$ $W$ , the largest possible value for $\newline$ $Y$ that is less than $\newline$ $W$ ( $\newline$ $9$ ) is $\newline$ $Y$ $0$ .
Identify Smallest Odd Digit for $X$ : Identify the smallest possible odd digit for $X$ to maximize the value of $W-X+Y-Z$ , considering that $WXYZ$ must be divisible by $X$ . $\newline$ Since $X$ must be odd and different from $W$ and $Y$ , the smallest possible value for $X$ is $1$ .
Determine Smallest Odd Digit for $Z$ : Determine the smallest possible odd digit for $Z$ to maximize the value of $W-X+Y-Z$ , considering that $WXYZ$ must be divisible by $Z$ . $\newline$ Since $Z$ must be odd and different from $W$ , $X$ , and $Y$ , the smallest possible value for $Z$ that is not $Z$ $0$ (already taken by $X$ ) is $Z$ $2$ .
Check Divisibility of $9713$ : Check if the number $9713$ is divisible by each of its digits. $\newline$ $9713$ is divisible by $9$ , $7$ , $1$ , and $3$ .
Calculate $W-X+Y-Z$ : Calculate the value of $W-X+Y-Z$ with the chosen digits. $\newline$ $W = 9$ , $X = 1$ , $Y = 7$ , $Z = 3$ $\newline$ $W-X+Y-Z = 9 - 1 + 7 - 3$ $\newline$ $W-X+Y-Z = 8 + 4$ $\newline$ $W-X+Y-Z = 12$

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