Answer:
a) Q = 0 or 9
b) Q = 5
c) Q = 7
d) Q = 8
Step-by-step explanation:
We assume the postal money order to be of the United State. A smudged digit is calculated using the following algorithm.
1. Add first 10 digits of 11-digit number.
2. Divide the sum of the 10 numbers by 9.
3.The remainder is the smudged digit.
4.The smudged digit is appended to the end of the ID number or anywhere in the number.
It is calculated as:
[tex]x_{11} = (x_{1} + x_{2} + x_{3} + x_{4} + x_{5} + x_{6} + x_{7} + x_{8} + x_{9} + x_{10} )mod 9[/tex]
a) 493212Q0688
[tex]8 = (4 + 9 + 3 + 2 + 1 + 2 + Q + 0 + 6 + 8) mod 9\\8 = (Q + 35) mod 9\\8 = Q mod 9 + 35 mod 9\\8 = Q mod 9 + 8\\Q mod 9 = 8 - 8\\Q mod 9 = 0\\Q = 0 or 9[/tex]
Therefore, Q is either 0 or 9.
b) 850Q9103858
[tex]8 = (8 + 5 + 0 + Q + 9 + 1 + 0 + 3 + 8 + 5) mod 9\\8 = (Q + 39) mod 9\\8 = Q mod 9 + 39 mod 9\\8 = Q mod 9 + 3\\Q mod 9 = 8 - 3\\Q mod 9 = 5\\Q = 5[/tex]
Therefore, Q is 5 because from Q mod 9 = 5, we have Q = 5 mod 9.
c) 2Q941007734
[tex]4 = (2 + Q + 9 + 4 + 1 + 0 + 0 + 7 + 7 + 3) mod 9\\4 = (Q + 33) mod 9\\4 = Q mod 9 + 33 mod 9\\4 = Q mod 9 + 6\\Q mod 9 = 4 - 6\\Q mod 9 = -2\\Q = -2 mod 9\\Q = 7[/tex]
Therefore, Q is 7 because we have to cancel out the negative sign by adding the modulo base (9) to the negative number: -2 + 9 = 7.
d) 66687Q03201
[tex]1 = (6 + 6 + 6 + 8 + 7 + Q + 0 + 3 + 2 + 0) mod 9\\1 = (Q + 38) mod 9\\1 = Q mod 9 + 38 mod 9\\1 = Q mod 9 + 2\\Q mod 9 = 1 - 2\\Q mod 9 = -1\\Q = -1 mod 9\\Q = 8[/tex]
Therefore, Q is 8 because we have to cancel out the negative sign by adding the modulo base (9) to the negative number: -1 + 9 = 8.
