Question

The United States Postal Service (USPS) sells money orders identified by an \(11\)-digit number\({x_1}{x_2} \ldots {x_{11}}.\) The first ten digits identify the money order;\({x_{11}}\)is a check digit that satisfies\({x_{11}} = {x_1} + {x_2} + L + {x_{10}}\,mod\,9\) Determine whether each of these numbers is a valid USPS money order identification number.

\(\begin{aligned}{*{20}{l}}{a) 74051489623}\\{b) 88382013445}\\{c) 56152240784}\\{d) 66606631178}\end{aligned}\)

Short answer

USPS identification number

\(74051489623\)is not valid

\(88382013445\)is valid

\(56152240784\)is valid

\(66606631178\)is not valid

Step-by-step solution

Step 1

For a USPS code\({x_1}{x_2} \ldots {x_{11}}\), the check digit is calculated using the following expression:

Step 2

(a)Given

\(7405189623\)

Use the above expression to calculate the check digits

\({x_{11}} = {x_1} + {x_2} + \ldots + {x_{10}}mod9\)

\(\begin{array}{l} = 7 + 4 + 0 + 5 + 1 + 4 + 8 + 9 + 6 + 2mod9\\ = 46mod9\\ = 1\end{array}\)

Thus, the check digit \(1\)

while the last digit of the given number is \(3\), Since these two digits are not the same, the USPS money order identification number is not valid.

Step 3

(b)Given

\(88382013445\)

Use the above expression to calculate the check digits:

\(\begin{array}{l}{x_{11}} = {x_1} + {x_2} + \ldots + {x_{10}}mod9\\ = 8 + 8 + 3 + 8 + 2 + 0 + 1 + 3 + 4 + 4mod9\\ = 41mod9\\ = 5\end{array}\)

Thus, the check digit is \(5\)

The check digit is \(5\), while the last digit of the given number is \(5\) . Since these two digits are the same, the USPS money order identification number is valid.

Step 4

(c)Given

\(56152240784\)

Use the above expression to calculate the check digit:

\({x_{11}} = {x_1} + {x_2} + \ldots + {x_{10}}mod9\)

\(\begin{array}{l} = 5 + 6 + 1 + 5 + 2 + 2 + 4 + 0 + 7 + 8mod9\\ = 40mod9\\ = 4\end{array}\)

The check digit is\(4\), while the last digit of the given number is\(4\) . Since these two digits are the same, the USPS money order identification number is valid.

Step 5

(d)Given

\(66606631178\)

\({x_{11}} = {x_1} + {x_2} + \ldots + {x_{10}}mod9\)

\(\begin{array}{l}6 + 6 + 6 + 0 + 6 + 6 + 3 + 1 + 1 + 7 + 8mod9\\ = 42mod9\\ = 6\end{array}\)

The check digit is \(6\) while the last digit of the given number is \(8\) . Since these two digits are not the same, the USPS money order identification number is not valid.