Question

Let us assign numeric values to the uppercase alphabet (A=0, B=1, . . , Z=25). We can do
modular arithmetic on the system using modulo 26
(i) What is (A+6) mod 26 in this system?
(ii) What is (C-10) mod 26 in this system?

Answer 1

Answer:

as usual, and then the remainder calculated at the end. Alternately, remainders can be calculated at any time

during the calculations, if they make the working easier.

Here are some examples to illustrate how modular addition, subtraction and multiplication work.

Example (modulo 26) Calculations

12 + 5 ≡ 17

25 + 1 ≡ 0 25 + 1 = 26 = 1 × 26 + 0

2 − 5 ≡ 23 2 − 5 = −3 = −1 × 26 + 23

38 + 109 ≡ 12 + 5 ≡ 17 109 = 4 × 26 + 5

46 + 32 ≡ 20 + 6 ≡ 26 ≡ 0 26 = 1 × 26 + 0

2 × 12 ≡ 24 24 = 0 × 26 + 24

21 × 6 = 126 ≡ 22 126 = 4 × 26 + 22

30 × 27 = 810 ≡ 4 810 = 31 × 26 + 4

30 × 27 ≡ 4 × 1 = 4 30 ≡ 4 and 27 ≡ 1

Another important concept to consider when working modulo n is the multiplicative inverse of a given

integer. For an integer x, its multiplicative inverse modulo n (if one exists), denoted x

−1

, is the number such

that x × x

−1 ≡ 1 modulo n. For example, the multiplicative inverse of 5 modulo 26 is 21, because 5 × 21 ≡ 1

modulo 26 (because 5 × 21 = 105 = 4 × 26 + 1 ≡ 1 modulo 26).

(It is important to note that in modular arithmetic, a

−1 does not mean 1/a. In fact, we have not defined

division at all.)

Not all numbers have a multiplicative inverse modulo n. In general, a number will only have an inverse if

it does not share any common factors with the modulus n (apart from the common factor 1). Since 26 has the

factors 2 and 13, this means that even numbers, and the number 13, do not have an inverse modulo 26.

Multiplicative inverses are useful in solving for x equations of the form a×x ≡ b modulo n. First, calculate

the inverse of a, if it exists, and then multiply both sides of the equation by a

−1

, giving a

−1 × a × x = 1 × x ≡

a

−1 × b modulo n. For example, to solve 23 × x ≡ 2 modulo 26, we proceed as follows. First, note that the

multiplicative inverse of 23 is 17 (mod 26), because 23 × 17 = 390 = 26 × 15 + 1 ≡ 1. Then,

since 23−1 ≡ 17 modulo 26,

23 × x ≡ 2 modulo 26 means that

17 × 23 × x ≡ 17 × 2 modulo 26.

Then since 17 × 23 ≡ 1 modulo 26,

x ≡ 34 modulo 26, so

x ≡ 8 modulo 26.

Answer 2

i) (A+6) mod 26 = 8 equivalent to 'G'.

ii) (C-10) mod 26 = 18 equivalent to 'S'.

Modular Arithmetic :

• Modulo in mathematics is the remainder obtained when a number is divided by the other
• It is denoted using the percent symbol i.e. "%"
• Thus, for two numbers 'a' and 'b', 'a mod b' denotes the integer remainder obtained when dividing a by b.
• Any number modulo 'N' can result in N values i.e. 0 to N-1.

i) The result is 6 which corresponds to 'G' in the given system

According to the given, the corresponding number for the character A is 0.

Thus,

        (A+6) mod 26 = (0+6) mod 26

                                = 6 mod 26

                                = remainder on dividing 6 by 26

                                = 6

Thus, the result is 6 which matches to the given system to the letter 'G'.

ii) The result is 18 which corresponds to 'S' in the given system

According to the given, the corresponding number for the character Cis 2.

Thus,

        (C-10) mod 26 = (2-10) mod 26

                                = (-8) mod 26

According to a rule in Modular Arithmetic for negative numbers :

            $(a)mod(b) = (a+b)mod(b)$

So, adding the divisor to the dividend will not modify the remainder. So, here we can add 26 to make the dividend positive.

                                =  (-8) mod 26

                                = (-8+26) mod 26

                                = 18 mod 26

                                = 18

Thus, the result is 18 which matches the given system to the letter 'S'.

To learn more about Modular Arithmetic, visit

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