A Universal Product Code $($UPC$)$ is a numeric code assigned to virtually all products

sold in stores; you usually see it as a bar code, which you scan at the cash register. The

UPC is a $12-$digit numeric code a$1...$ a$12$; the $12$ digits have to satisfy the condition

below:

$3$a$1+$ a$2+3$a$3+$ a$4++$ a$120$ mod $10$

The sum $3$a$1+$ a$2+3$a$3+$ a$4++$ a$12$ mod $10$ is called the checksum.

If the digits do not satisfy this condition, it can not be a valid UPC code $$ so

probably there was an error when scanning the barcode and it needs to be re$-$scanned.

For example, $380177-051136$ is a valid UPC code $($check$!)$; however, if the first $3$ was

replaced by $8,$ this would change the checksum by $3\backslash$times $5=155$ mod $10,$ so the

new checksum woudln$$t be zero.

Show that this scheme detects all single$-$digit errors. Does it detect all transpositions

$($i$.$e$.$ interchange of two adjacent digits$)?$