Problem #1 ZIP Code: Given the following ZIP code (4X122Y), which corresponds to

Question

Problem #1 ZIP Code: Given the following ZIP code (4X122Y), which corresponds to an address in the fourth zone.

(a) What is the set of all possible values for the check digit Y? For more details about the zones defined in the US, you may refer to your lectures.

(b) Assume that the USPS office would like to add 4-digit extension to the ZIP code mentioned above. Give three possible sets of such 4-digit extension such that your answer for (a) will not change.

Problem #2 Given a credit card number (5518-X201-2345-6781),

(a) What is the suitable value for X?

(b) If we would like to make the check digit 5, what is the suitable value for X?

Problem #3 The routing number of a bank in the U.S. is (123271978). For some reasons, the bank is asked to eliminate ‘7’ from the routing number. Notice that ‘7’ appears twice in the routing number. Provide four possible options to replace ‘7’ in the routing number without changing the others digits in the routing number. For example, one option is (4,0). So if we replace the first ‘7’ by ‘4’and the second ‘7’ by ‘0’, the result is (123241908). The later is still a valid routing number and we did no change to other digits to the other digits.

Problem #4 Assume an ISBN (87 - 11 - 07559 – 7) is proposed for a new book. The publisher does not want ‘5’ to appear in the ISBN. Without changing other digits in the ISBN, provide four options to replace ‘5’, which appears twice in the ISBN. For example, (8,1) is an option. This is because (87 - 11 - 07819 – 7) is a valid ISBN.

Explanation / Answer

Concept behind answering these questions:

The validity of these numbers like postal code, credit card are examined by mod10, mod9, or mod11 rules. Here, we are checking these numbers correctness by mod 10 rule i.e. Luhn algorithm. The last digit of given number is called the check digit. To, validate follow these steps:

i) Multiply each second digit from right most end.

ii) If the product is > 9, then subtract 9 from it.

iii) Find the sum of these numbers

iv) If sum modulo 10 is equal to check digit, then number is valid.

For example, take number 456231. Here, check digit is 1.

Step 1 : 8 5 12 2 6 1 (doubling each second digit from right end)

Step 2 : 8 5 3 2 6 1 (if product is >9, subtract 9 from the product)

Step 3 : 8+5+3+2+6+1 = 25

Step 4 : 25 modulo 10 = 5 which is not equal to 1. So, this number is not valid by rules.

Now coming back to given questions:

1)

a) ZIP Code : 4X122Y

Step 1 : 8 X 2 2 4 Y (doubling each second digit from right end)

Step 2 : 8 X 2 2 4 Y (if product is >9, subtract 9 from the product)

Step 3 : 8+X+2+2+4+Y = 16+X+Y

Step 4 : For this ZIP to be valid, (16+X+Y) mod 10 should be equal to Y. Now, X can be from 0 to 9 so, 16+X must be perfect multiple of 10. The possible values for X is 4. Y can take any value from 0 to 9.

b) Let the four digit extension be abcd, for it to be valid, 2a+b+2c+d must be perfect multiple of 10. Thus, 3 such sets are: 1313, 3111, 1222

2) Credit card number : 5518-X201-2345-6781

a)

Step 1 : 10 5 2 8 2X 2 0 1 4 3 8 5 12 7 16 1 (doubling each second digit from right end)

Step 2 :   1 5 2 8 (2X modulo 9) 2 0 1 4 3 8 5 3 7 7 1 (if product is >9, subtract 9 from the product)

Step 3 : 57+ (2X mod 9)

Step 4 : check digit is 1. So, the sum can be 61. Therefore, (2X) = 4. Therefore, X = 2.

b) If check digit is 5, then new sum will be 57-1+5+(2X mod 9). Thus, sum should be 65. So, 2X = 4, X = 2.

3) 123271978

Let us replace 7 with X and Y. SO, the number is 1232X19Y8.

Sum is 1+ 4+ 3+4+X+2+9+(2Y mod 9)+8 = 31+X+(2Y mod 9). Since check digit is 8, so sum should be 38 or 48, X+(2Y mod 9) = 7 or 17.

So, possible combination for (X,Y) are (0,4), (7,0), (3, 2), (1, 3), (5, 1).

4. ISBN : 87 - 11 - 07559 – 7

Replacing (5, 5) with (X,Y), number becomes 871107XY97

So, sum is 40 + 2X mod 9 +Y. Check digit is 7.

Thus, possible equations are 2X + Y = 7 or 17 or 16 or 26.

Thus possible combinations are (0,7), (9, 8), (8, 0), (3, 1), (5, 7) by satisfying one of the above equation.

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