This question's explanation and your answers so far.
問97
頻出度★★☆
多くの市販の書籍には，書籍を識別するための ISBN（International Standard Book Number）コードが付けられている。ISBN コードは，0～9の数字を使った13桁の記号で構成され，左側から桁を数える。最も左側の桁を1桁目とする。1桁目から12桁目までは，国記号，出版者記号及び書籍固有の記号などが含まれる。
ISBN コードの13桁目（最も右側の桁）はチェック数字と呼ばれる桁である。ISBN コードのチェック数字は，1桁目から12桁目の数字を使って，次の計算方法によって求める。ここで，図1のように1桁目から12桁目の数字を a₁～a₁₂，チェック数字を X とする。
〔チェック数字の計算方法〕
(1) a₁～a₁₂について，奇数桁には1，偶数桁には3の重みを付ける。
(2) a₁～a₁₂の各桁に，それぞれの桁に対応した重みを掛けたものの和 S を求める。
S ＝ a₁×1＋a₂×3＋a₃×1＋…＋a₁₂×3
(3) S を10で割った余り R を求める。
(4) 10から R を引いた値をチェック数字 X とする。なお，この値が10になった場合は，チェック数字は0とする。
ISBN コードの1桁目から12桁目が次の数字の場合, 13桁目に付加されるチェック数字 X はどれか。
桁
1
2
3
4
5
6
7
8
9
10
11
12
13
9
7
8
4
0
0
0
0
8
6
0
1
X
出典 平成24年度 春期 ITパスポート試験 問97
解説
問 97
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○ア
正解The correct answer is ア (X = 1). The check digit of an ISBN code (13 digits) is obtained with the same “modulus 10, weight 3” method used for JAN codes. The steps are as follows. (1) Starting from the 1st digit, multiply each digit by the weights 1, 3, 1, 3, … (odd-numbered positions ×1, even-numbered positions ×3). (2) Add up the 12 products. (3) Subtract the remainder of that total divided by 10 from 10 (if the remainder is 0, the check digit is also 0). Using the digits in the figure: 9×1=9, 7×3=21, 8×1=8, 4×3=12, 0×1=0, 0×3=0, 0×1=0, 0×3=0, 8×1=8, 6×3=18, 0×1=0, 1×3=3, so the total is 9+21+8+12+8+18+3 = 79. The remainder of 79 divided by 10 is 9, so the check digit is 10 − 9 = 1. As a verification, simply confirm that the weighted total of all 13 digits including the check digit, 79 + 1 = 80, is exactly a multiple of 10.
×Incorrectイ
3 is the value obtained by adding the 12 digits without any weighting, 9+7+8+4+0+0+0+0+8+6+0+1 = 43, and then taking its ones digit directly as the check digit. The check digit must use the total after multiplying by the weights 1 and 3, and it is not the remainder itself but the value of 10 minus the remainder. Adding 3 to the correct weighted total 79 gives only 82, which is not a multiple of 10, so it does not satisfy the condition.
×Incorrectウ
7 is the value that appears when the order of the weights is mistaken and the digits are multiplied by 3, 1, 3, 1, … starting from the 1st digit (total 93 → 10 − 3 = 7), or when 10 − 3 = 7 is taken from the unweighted simple total 43. In ISBN-13 the weights always start with 1 for the 1st digit and 3 for the 2nd digit, so this is wrong. 79 + 7 = 86 is not a multiple of 10 either.
×Incorrectエ
9 is the value obtained by taking the remainder 9 of the weighted total 79 divided by 10 directly as the check digit. In modulus 10 weight 3, what is appended is not the remainder but the value of 10 minus the remainder. Since 79 + 9 = 88 is not a multiple of 10, it cannot serve as the check digit.
ここがポイント
The check digit of ISBN-13 (the same system as JAN/EAN-13) is calculated with modulus 10, weight 3. Divide by 10 the total obtained by multiplying the digits by the weights 1, 3, 1, 3, … from the 1st digit onward, and the value of 10 minus the remainder (0 if the remainder is 0) becomes the final digit.
A check digit is a redundant digit added in order to detect key-entry mistakes and reading errors, and it is defined so that the weighted total including the check digit becomes exactly a multiple of 10. This property can be used to verify the calculation.