{"id":3223,"date":"2025-06-09T08:35:15","date_gmt":"2025-06-09T08:35:15","guid":{"rendered":"https:\/\/diznr.com\/?p=3223"},"modified":"2025-06-09T08:35:15","modified_gmt":"2025-06-09T08:35:15","slug":"day-06part-09-operating-system-for-gate-arithmetic-modulo-additive-and-modulo-multiplicative","status":"publish","type":"post","link":"https:\/\/www.reilsolar.com\/pdf\/day-06part-09-operating-system-for-gate-arithmetic-modulo-additive-and-modulo-multiplicative\/","title":{"rendered":"Day 06Part 09- Operating System for gate- Arithmetic Modulo- Additive and Multiplicative Modulo"},"content":{"rendered":"

Day 06Part 09- Operating System for gate- Arithmetic Modulo- Additive and Multiplicative Modulo<\/p>\n

[fvplayer id=”306″]<\/p>\n

Day 06 Part 09 – Operating System for GATE<\/strong><\/h3>\n

\u00a0Arithmetic Modulo – Additive and Multiplicative Modulo<\/strong><\/h3>\n

\u00a0What is Modulo Arithmetic?<\/strong><\/h3>\n

Modulo Arithmetic<\/strong> (also called mod arithmetic<\/strong>) deals with remainders<\/strong> when numbers are divided. It is used in cryptography, hashing, clock arithmetic, and operating systems<\/strong>.<\/p>\n

The modulo operation is written as:<\/p>\n

Amod\u2009\u2009BA \\mod B<\/span>A<\/span><\/span>mod<\/span><\/span>B<\/span><\/span><\/span><\/span><\/span><\/p>\n

This gives the remainder when A is divided by B<\/strong>.<\/p>\n

Example:<\/strong><\/p>\n

17mod\u2009\u20095=217 \\mod 5 = 2<\/span>17<\/span><\/span>mod<\/span><\/span>5<\/span>=<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/p>\n

(Since 17\u00f75=317 \\div 5 = 3<\/span>17<\/span>\u00f7<\/span><\/span>5<\/span>=<\/span><\/span>3<\/span><\/span><\/span><\/span> remainder 2<\/strong>).<\/p>\n

\u00a0Additive Modulo<\/strong><\/h3>\n

If we add two numbers under a modulo, we use the formula:<\/p>\n

(A+B)mod\u2009\u2009M=((Amod\u2009\u2009M)+(Bmod\u2009\u2009M))mod\u2009\u2009M(A + B) \\mod M = ((A \\mod M) + (B \\mod M)) \\mod M<\/span>(<\/span>A<\/span>+<\/span><\/span>B<\/span>)<\/span><\/span>mod<\/span><\/span>M<\/span>=<\/span><\/span>((<\/span>A<\/span><\/span>mod<\/span><\/span>M<\/span>)<\/span>+<\/span><\/span>(<\/span>B<\/span><\/span>mod<\/span><\/span>M<\/span>))<\/span><\/span>mod<\/span><\/span>M<\/span><\/span><\/span><\/span><\/span><\/p>\n

Example:<\/strong><\/p>\n

(7+5)mod\u2009\u20094(7 + 5) \\mod 4<\/span>(<\/span>7<\/span>+<\/span><\/span>5<\/span>)<\/span><\/span>mod<\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/p>\n

Step 1: 7mod\u2009\u20094=37 \\mod 4 = 3<\/span>7<\/span><\/span>mod<\/span><\/span>4<\/span>=<\/span><\/span>3<\/span><\/span><\/span><\/span>, 5mod\u2009\u20094=15 \\mod 4 = 1<\/span>5<\/span><\/span>mod<\/span><\/span>4<\/span>=<\/span><\/span>1<\/span><\/span><\/span><\/span>
Step 2: (3+1)mod\u2009\u20094=4mod\u2009\u20094=0(3 + 1) \\mod 4 = 4 \\mod 4 = 0<\/span>(<\/span>3<\/span>+<\/span><\/span>1<\/span>)<\/span><\/span>mod<\/span><\/span>4<\/span>=<\/span><\/span>4<\/span><\/span>mod<\/span><\/span>4<\/span>=<\/span><\/span>0<\/span><\/span><\/span><\/span>
\u00a0Answer: 0<\/strong><\/p>\n

\u00a0Multiplicative Modulo<\/strong><\/h3>\n

If we multiply two numbers under a modulo, we use the formula:<\/p>\n

(A\u00d7B)mod\u2009\u2009M=((Amod\u2009\u2009M)\u00d7(Bmod\u2009\u2009M))mod\u2009\u2009M(A \\times B) \\mod M = ((A \\mod M) \\times (B \\mod M)) \\mod M<\/span>(<\/span>A<\/span>\u00d7<\/span><\/span>B<\/span>)<\/span><\/span>mod<\/span><\/span>M<\/span>=<\/span><\/span>((<\/span>A<\/span><\/span>mod<\/span><\/span>M<\/span>)<\/span>\u00d7<\/span><\/span>(<\/span>B<\/span><\/span>mod<\/span><\/span>M<\/span>))<\/span><\/span>mod<\/span><\/span>M<\/span><\/span><\/span><\/span><\/span><\/p>\n

Example:<\/strong><\/p>\n

(7\u00d75)mod\u2009\u20094(7 \\times 5) \\mod 4<\/span>(<\/span>7<\/span>\u00d7<\/span><\/span>5<\/span>)<\/span><\/span>mod<\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/p>\n

Step 1: 7mod\u2009\u20094=37 \\mod 4 = 3<\/span>7<\/span><\/span>mod<\/span><\/span>4<\/span>=<\/span><\/span>3<\/span><\/span><\/span><\/span>, 5mod\u2009\u20094=15 \\mod 4 = 1<\/span>5<\/span><\/span>mod<\/span><\/span>4<\/span>=<\/span><\/span>1<\/span><\/span><\/span><\/span>
Step 2: (3\u00d71)mod\u2009\u20094=3mod\u2009\u20094=3(3 \\times 1) \\mod 4 = 3 \\mod 4 = 3<\/span>(<\/span>3<\/span>\u00d7<\/span><\/span>1<\/span>)<\/span><\/span>mod<\/span><\/span>4<\/span>=<\/span><\/span>3<\/span><\/span>mod<\/span><\/span>4<\/span>=<\/span><\/span>3<\/span><\/span><\/span><\/span>
\u00a0Answer: 3<\/strong><\/p>\n

\u00a0Applications in Operating Systems & GATE<\/strong><\/h3>\n

Hashing Functions<\/strong> (Used in databases & OS)
Clock Arithmetic<\/strong> (Example: 12-hour clock)
Encryption & Cryptography<\/strong> (RSA Algorithm)
Scheduling Algorithms<\/strong> (Round-Robin Scheduling)<\/p>\n

Would you like practice questions or code examples in Python\/C++?<\/strong><\/p>\n

Day 06Part 09- Operating System for gate- Arithmetic Modulo- Additive and Multiplicative Modulo<\/a><\/h3>\n

UNIT-II Modular Arithmetic and Cryptography<\/a><\/h3>\n

UNIT- III Modular Arithmetic<\/a><\/h3>\n

Here is a concise and exam-oriented explanation<\/strong> of Arithmetic Modulo Operations<\/strong> \u2013 specifically Additive and Multiplicative Modulo<\/strong>, perfect for Day 06 \u2013 Part 09<\/strong> of an Operating System or GATE Computer Science<\/strong> preparation series.<\/p>\n

\n

\ud83d\udcd8 Day 06 \u2013 Part 09: Arithmetic Modulo \u2013 Additive & Multiplicative Modulo<\/strong><\/h2>\n
\n

\ud83d\udd39 What is Modulo Arithmetic?<\/strong><\/h2>\n

Modulo arithmetic deals with the remainder<\/strong> when one number is divided by another.
\nIt\u2019s widely used in computer science<\/strong>, especially in memory management, hash functions, cyclic counters, cryptography<\/strong>, and operating systems<\/strong>.<\/p>\n

\n

\ud83d\udd22 General Definition:<\/strong><\/h3>\n

a\u200amod\u200am=remainder\u00a0when\u00a0a\u00a0is\u00a0divided\u00a0by\u00a0ma \\bmod m = \\text{remainder when } a \\text{ is divided by } m<\/span><\/p>\n

Example:<\/p>\n

17\u200amod\u200a5=2(since\u00a017\u00a0=\u00a03\u00a0\u00d7\u00a05\u00a0+\u00a02)17 \\bmod 5 = 2 \\quad \\text{(since 17 = 3 \u00d7 5 + 2)}<\/span><\/p>\n

\n

\ud83e\uddee 1. Additive Modulo<\/strong><\/h2>\n

\u2705 Rule:<\/strong><\/h3>\n

(a+b)mod\u2009\u2009m=((amod\u2009\u2009m)+(bmod\u2009\u2009m))mod\u2009\u2009m(a + b) \\mod m = \\left( (a \\mod m) + (b \\mod m) \\right) \\mod m<\/span><\/p>\n

\ud83d\udd0d Example:<\/h3>\n

Let a=14a = 14<\/span>, b=9b = 9<\/span>, and m=5m = 5<\/span><\/p>\n

(14+9)mod\u2009\u20095=23mod\u2009\u20095=3(14 + 9) \\mod 5 = 23 \\mod 5 = 3<\/span><\/p>\n

Using the rule:<\/p>\n

(14mod\u2009\u20095)=4,(9mod\u2009\u20095)=4\u21d2(4+4)mod\u2009\u20095=8mod\u2009\u20095=3\u2705(14 \\mod 5) = 4,\\quad (9 \\mod 5) = 4 \\Rightarrow (4 + 4) \\mod 5 = 8 \\mod 5 = 3 \u2705<\/span><\/p>\n

\n

\u2716\ufe0f 2. Multiplicative Modulo<\/strong><\/h2>\n

\u2705 Rule:<\/strong><\/h3>\n

(a\u00d7b)mod\u2009\u2009m=((amod\u2009\u2009m)\u00d7(bmod\u2009\u2009m))mod\u2009\u2009m(a \\times b) \\mod m = \\left( (a \\mod m) \\times (b \\mod m) \\right) \\mod m<\/span><\/p>\n

\ud83d\udd0d Example:<\/h3>\n

Let a=7a = 7<\/span>, b=6b = 6<\/span>, and m=5m = 5<\/span><\/p>\n

(7\u00d76)mod\u2009\u20095=42mod\u2009\u20095=2(7 \u00d7 6) \\mod 5 = 42 \\mod 5 = 2<\/span><\/p>\n

Using the rule:<\/p>\n

(7mod\u2009\u20095)=2,(6mod\u2009\u20095)=1\u21d2(2\u00d71)mod\u2009\u20095=2\u2705(7 \\mod 5) = 2,\\quad (6 \\mod 5) = 1 \\Rightarrow (2 \u00d7 1) \\mod 5 = 2 \u2705<\/span><\/p>\n

\n

\ud83e\udde0 Why Modulo is Useful in OS & GATE?<\/strong><\/h2>\n\n\n\n\n\n\n\n\n\n
Area<\/th>\nUse of Modulo<\/th>\n<\/tr>\n<\/thead>\n
Round Robin Scheduling<\/td>\nTo wrap process index in a circular queue<\/td>\n<\/tr>\n
Page Replacement<\/td>\nHashing page IDs<\/td>\n<\/tr>\n
Hash Tables<\/td>\nDistribute keys uniformly<\/td>\n<\/tr>\n
Addressing<\/td>\nCalculate cyclic buffers or indices<\/td>\n<\/tr>\n
Cryptography<\/td>\nModular arithmetic in encryption algorithms<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
\n

\ud83d\udccc Properties of Modulo Arithmetic<\/strong><\/h2>\n\n\n\n\n\n\n\n
Operation<\/th>\nModulo Rule<\/th>\n<\/tr>\n<\/thead>\n
Addition<\/td>\n(a+b)mod\u2009\u2009m=((amod\u2009\u2009m)+(bmod\u2009\u2009m))mod\u2009\u2009m(a + b) \\mod m = ((a \\mod m) + (b \\mod m)) \\mod m<\/span><\/td>\n<\/tr>\n
Multiplication<\/td>\n(a\u22c5b)mod\u2009\u2009m=((amod\u2009\u2009m)\u22c5(bmod\u2009\u2009m))mod\u2009\u2009m(a \\cdot b) \\mod m = ((a \\mod m) \\cdot (b \\mod m)) \\mod m<\/span><\/td>\n<\/tr>\n
Subtraction<\/td>\n(a\u2212b)mod\u2009\u2009m=((amod\u2009\u2009m)\u2212(bmod\u2009\u2009m)+m)mod\u2009\u2009m(a – b) \\mod m = ((a \\mod m) – (b \\mod m) + m) \\mod m<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
\n

\ud83d\udcda GATE-Level Question Example:<\/strong><\/h2>\n

Q:<\/strong>
\nWhat is the value of:<\/p>\n

((27+35)\u00d79)mod\u2009\u20097((27 + 35) \\times 9) \\mod 7<\/span><\/p>\n

Step-by-step:<\/strong><\/p>\n

    \n
  1. (27+35)=62(27 + 35) = 62<\/span><\/li>\n
  2. 62\u00d79=55862 \u00d7 9 = 558<\/span><\/li>\n
  3. 558mod\u2009\u20097=6558 \\mod 7 = 6<\/span><\/li>\n<\/ol>\n

    \u2705 Answer: 6<\/strong><\/p>\n

    \n

    \ud83d\udcd6 Summary<\/h2>\n\n\n\n\n\n\n\n\n
    Topic<\/th>\nKey Point<\/th>\n<\/tr>\n<\/thead>\n
    Additive Modulo<\/td>\nWorks by summing remainders<\/td>\n<\/tr>\n
    Multiplicative Modulo<\/td>\nWorks by multiplying remainders<\/td>\n<\/tr>\n
    Common Uses<\/td>\nScheduling, hashing, cyclic buffers<\/td>\n<\/tr>\n
    Formulae<\/td>\n(a\u00b1b)mod\u2009\u2009m(a \\pm b) \\mod m<\/span>, (a\u22c5b)mod\u2009\u2009m(a \\cdot b) \\mod m<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
    \n

    Would you like:<\/p>\n