Modular Subtraction

Modular Subtraction#

Modular subtraction involves subtracting one number from another and then taking the remainder when the result is divided by the modulus.

The formula for modular subtraction of two numbers \(a\) and \(b\) with modulus \(m\) is:

\[ (a - b) \mod m \]

Steps:

  1. Subtract the number \(b\) from \(a\).

  2. If the result is negative, add the modulus \(m\) until the result is non-negative.

  3. Apply the modulus \(m\) to find the remainder.

Functions#

Hide code cell source 
import sympy as sp
from sympy import *
from math import gcd
import random

def modular_subtraction(a, b, modulus):
    result = (a - b) % modulus
    print(f"{a} - {b} ≡ {result} (mod {modulus})")
    return result

Examples#

Finding \((10 - 17)~mod~7\)#

a = 10
b = 17
modulus = 7

result = modular_subtraction(a, b, modulus)

10 - 17 ≡ 0 (mod 7)

Explanation

To find the result of ( 10 - 17 ) modulo ( 7 ), follow these steps:

  1. Perform the subtraction:

\[ 10 - 17 = -7 \]

  1. Adjust to ensure the result is within the range 0 to modulus-1:

Since we have a negative result, we adjust it by adding the modulus until it becomes non-negative:

\[ -7 + 7 = 0 \]

  1. The adjusted result is the result of the modulo operation:

\[ 10 - 17 \equiv 0 \mod 7 \]

Finding \((217 - 240)~mod~19\)#

a = 217
b = 240
modulus = 19

result = modular_subtraction(a, b, modulus)

217 - 240 ≡ 15 (mod 19)

Explanation

To find the result of ( 217 - 240 ) modulo ( 19 ), follow these steps:

  1. Perform the subtraction:

\[ 217 - 240 = -23 \]

  1. Adjust to ensure the result is within the range 0 to modulus-1:

Since we have a negative result, we adjust it by adding the modulus until it becomes non-negative:

\[ -23 + 19 = -4 \]

  1. Find the remainder when divided by the modulus:

Now, calculate the remainder when ( -4 ) is divided by ( 19 ):

\[ -4 \mod 19 = 15 \]

Therefore:

\[ 217 - 240 \equiv 15 \mod 19 \]

Finding \((2293 - 334)~mod~38\)#

a = 2293
b = 334
modulus = 38

result = modular_subtraction(a, b, modulus)

2293 - 334 ≡ 21 (mod 38)

Explanation

To find the result of ( 2293 - 334 ) modulo ( 38 ), follow these steps:

  1. Perform the subtraction:

\[ 2293 - 334 = 1959 \]

  1. Find the remainder when divided by the modulus:

Now, calculate the remainder when ( 1959 ) is divided by ( 38 ):

\[ 1959 \div 38 = 51 \text{ remainder } 21 \]

  1. Express the division:

\[ 1959 = 38 \times 51 + 21 \]

  1. The remainder is the result of the modulo operation:

\[ 1959 \mod 38 = 21 \]

Therefore:

\[ 2293 - 334 \equiv 21 \mod 38 \]
Contents