Mathematics in Action: Vehicle Identification Numbers

Motor vehicles have a 17-character Vehicle Identification Number or VIN on a metal plate like the one below, usually on the driver’s side dashboard, or on the driver’s side door jamb, or in front of the engine block:


A Vehicle Identification Number (VIN) plate (Photo: Michiel1972)

VINs offer an interesting example of check digit calculation. The central digit (or an X representing 10) is a check digit (calculated modulo 11) used to detect errors. Any letters in the rest of the VIN are decoded like this:

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
1 2 3 4 5 6 7 8 1 2 3 4 5 7 9 2 3 4 5 6 7 8 9

The check digit calculation involves decoding the VIN, and multiplying the resulting numbers by the weights shown in blue, giving the products in purple:

VIN L J C P C B L C X 1 1 0 0 0 2 3 7
Decoded 3 1 3 7 3 2 3 3 10 1 1 0 0 0 2 3 7
Weights 8 7 6 5 4 3 2 10 0 9 8 7 6 5 4 3 2
Product 24 7 18 35 12 6 6 30 0 9 8 0 0 0 8 9 14

These products are added up modulo 11 (meaning the sum is divided by 11 and the remainder taken). In this case, the sum is 186 = 10 = X (mod 11), which makes the VIN valid, because it matches the original central X. What about the VIN on your vehicle?