Chapter 4: Q45SE (page 307)

Show that if $d_{1} d_{2} \dots d_{9}$ is a valid RTN, then $d_{9} = 7 (d_{1} + d_{4} + d_{7}) + 3 (d_{2} + d_{5} + d_{8}) + 9 (d_{3} + d_{6})$ mod 10Furthermore, use this formula to find the check digit that follows the eight digits 11100002 in a valid RTN.

Short Answer

Expert verified

$d_{9} = 7 (d_{1} + d_{4} + d_{7}) + 3 (d_{2} + d_{5} + d_{8}) + 9 (d_{3} + d_{6})$

Check digit: 5

Step by step solution

Step 1

For a routing transit number $d_{1} d_{2} \dots d_{9}$ the check digit is determined by the congruence:

$3 (d_{1} + d_{4} + d_{7}) + 7 (d_{2} + d_{5} + d_{8}) + {(d_{3} + d_{6} + d_{9})}^{\circ} 0 (mod 10)$

Step 2

To proof:

$d_{9} = 7 (d_{1} + d_{4} + d_{7}) + 3 (d_{2} + d_{5} + d_{8}) + 9 (d_{3} + d_{6})$

PROOF

From the given congruence, we obtain:

$3 (d_{1} + d_{4} + d_{7}) + 7 (d_{2} + d_{5} + d_{8}) + (d_{3} + d_{6} + d_{9}) mod 10 = 0 = 0 mod 10$

Subtract $3 (d_{1} + d_{4} + d_{7}) + 7 (d_{2} + d_{5} + d_{8})$ from each side of the equation:

$(d_{3} + d_{6} + d_{9}) mod 10 = - 3 (d_{1} + d_{4} + d_{7}) - 7 (d_{2} + d_{5} + d_{8}) mod 10$

Subtract $d_{3} + d_{6}$ from each side of the equation:

$d_{9} mod 10 = - 3 (d_{1} + d_{4} + d_{7}) - 7 (d_{2} + d_{5} + d_{8}) - d_{3} - d_{6} mod 10$

Since $- 3 mod 10 = 7, - 7 mod 10 = 3 and - 1 mod 10 = 9 ∣$

$d_{9} mod 10 = 7 (d_{1} + d_{4} + d_{7}) + 3 (d_{2} + d_{5} + d_{8}) + 9 d_{3} + 9 d_{6} mod 10$

Factor out the 9:

$d_{9} mod 10 = 7 (d_{1} + d_{4} + d_{7}) + 3 (d_{2} + d_{5} + d_{8}) + 9 (d_{3} + d_{6}) mod 10$

Step 3

DETERMINE CHECK DIGIT 1100002

Let us check that the required congruence for a ISBN-13 code is satisfied.

$\begin{aligned} d_{9} = 7 (d_{1} + d_{4} + d_{7}) + 3 (d_{2} + d_{5} + d_{8}) + 9 (d_{3} + d_{6}) mod 10 \\ = 7 (1 + 0 + 0) + 3 (1 + 0 + 2) + 9 (1 + 0) mod 10 \end{aligned}$

$\begin{aligned} = 7 (1) + 3 (3) + 9 (1) mod 10 \\ = 7 + 9 + 9 mod 10 \\ = 25 mod 10 \\ = 5 \end{aligned}$

Thus the check digit is then 5 .

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Show that if $d_{1} d_{2} \dots d_{9}$ is a valid RTN, then $d_{9} = 7 (d_{1} + d_{4} + d_{7}) + 3 (d_{2} + d_{5} + d_{8}) + 9 (d_{3} + d_{6})$ mod 10Furthermore, use this formula to find the check digit that follows the eight digits 11100002 in a valid RTN.

Short Answer

Step by step solution

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Step 2

Step 3

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Show that if d1d2…d9is a valid RTN, then d9=7(d1+d4+d7)+3(d2+d5+d8)+9(d3+d6)mod 10Furthermore, use this formula to find the check digit that follows the eight digits 11100002 in a valid RTN.

Short Answer

Step by step solution

Step 1

Step 2

Step 3

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Geometry

Logic and Functions

Probability and Statistics

Calculus

Decision Maths

Study anywhere. Anytime. Across all devices.

Show that if $d_{1} d_{2} \dots d_{9}$ is a valid RTN, then $d_{9} = 7 (d_{1} + d_{4} + d_{7}) + 3 (d_{2} + d_{5} + d_{8}) + 9 (d_{3} + d_{6})$ mod 10Furthermore, use this formula to find the check digit that follows the eight digits 11100002 in a valid RTN.