{"id":220203,"date":"2025-05-28T03:34:11","date_gmt":"2025-05-28T03:34:11","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=220203"},"modified":"2025-05-28T03:34:14","modified_gmt":"2025-05-28T03:34:14","slug":"multiplicative-inverse","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/05\/28\/multiplicative-inverse\/","title":{"rendered":"multiplicative inverse"},"content":{"rendered":"\n
\"\"<\/figure>\n\n\n\n

inverse in Zn? b) Use the definition of multiplicative inverse and trial and error to find the multiplicative inverse of 3 in Z16. Use the multiplicative inverse to solve 3x + 9 = 15 in Z16. c) Use the Pulverizer to find the multiplicative inverse of 37 in Z45 or explain why you cannot do so. If you find a multiplicative inverse, use it to solve 37x + 23 = 11 in Z45. d) Use the Pulverizer to find the multiplicative inverse of 39 in Z45 or explain why you cannot do so. If you find a multiplicative inverse, use it to solve 39x + 10 = 44. e) Use the Pulverizer to find the multiplicative inverse of 15 in Z26 or explain why you cannot do so. If you find a multiplicative inverse, use the multiplicative inverse to solve 15x + 5 = 23 in Z26. f) Use the Pulverizer to find the multiplicative inverse of 18 in Z32 or explain why you cannot do so. If you find a multiplicative inverse, use the multiplicative inverse to solve 18x + 6 = 30 in Z32
<\/p>\n\n\n\n

T he Correct Answer and Explanation is :<\/mark><\/strong><\/p>\n\n\n\n

.<\/p>\n\n\n\n

\n\n\n\n

a)<\/strong> Definition of a multiplicative inverse in $\\mathbb{Z}_n$:<\/h3>\n\n\n\n

An element $a \\in \\mathbb{Z}_n$ has a multiplicative inverse<\/strong> if there exists an integer $b \\in \\mathbb{Z}_n$ such that:<\/p>\n\n\n\n

$$
ab \\equiv 1 \\pmod{n}
$$<\/p>\n\n\n\n

This means $b$ is the inverse of $a$ modulo $n$, and $a \\cdot b \\mod n = 1$.<\/p>\n\n\n\n

\n\n\n\n

b)<\/strong> Trial and error to find the inverse of $3$ in $\\mathbb{Z}_{16}$:<\/h3>\n\n\n\n

Try values:<\/p>\n\n\n\n

    \n
  • $3 \\cdot 11 = 33 \\equiv 1 \\pmod{16}$<\/li>\n<\/ul>\n\n\n\n

    So, inverse of 3 is 11<\/strong>.<\/p>\n\n\n\n

    Now solve:<\/p>\n\n\n\n

    $$
    3x + 9 \\equiv 15 \\pmod{16} \\Rightarrow 3x \\equiv 6 \\pmod{16}
    \\Rightarrow x \\equiv 11 \\cdot 6 = 66 \\equiv \\boxed{2} \\pmod{16}
    $$<\/p>\n\n\n\n

    \n\n\n\n

    c)<\/strong> Pulverizer for inverse of 37 in $\\mathbb{Z}_{45}$:<\/h3>\n\n\n\n

    Use Extended Euclidean Algorithm:<\/p>\n\n\n\n

      \n
    • $\\gcd(37, 45) = 1$, so inverse exists.<\/li>\n\n\n\n
    • Solve $37x \\equiv 1 \\pmod{45}$
      Using Pulverizer:<\/li>\n<\/ul>\n\n\n\n

      $$
      1 = 7(45) – 8(37) \\Rightarrow -8 \\cdot 37 \\equiv 1 \\Rightarrow x \\equiv \\boxed{37} \\pmod{45}
      $$<\/p>\n\n\n\n

      Solve:<\/p>\n\n\n\n

      $$
      37x + 23 \\equiv 11 \\Rightarrow 37x \\equiv -12 \\equiv 33
      \\Rightarrow x \\equiv 37 \\cdot 33 = 1221 \\equiv \\boxed{6} \\pmod{45}
      $$<\/p>\n\n\n\n

      \n\n\n\n

      d)<\/strong> Inverse of 39 mod 45:<\/h3>\n\n\n\n

      $$
      \\gcd(39, 45) = 3 \\neq 1 \\Rightarrow \\text{No inverse.}
      $$<\/p>\n\n\n\n

      \n\n\n\n

      e)<\/strong> Inverse of 15 mod 26:<\/h3>\n\n\n\n

      $$
      \\gcd(15, 26) = 1 \\Rightarrow \\text{Inverse exists}
      \\Rightarrow \\boxed{7 \\cdot 15 = 105 \\equiv 1 \\pmod{26}}
      $$<\/p>\n\n\n\n

      Solve:<\/p>\n\n\n\n

      $$
      15x + 5 = 23 \\Rightarrow 15x = 18 \\Rightarrow x = 7 \\cdot 18 = 126 \\equiv \\boxed{22} \\pmod{26}
      $$<\/p>\n\n\n\n

      \n\n\n\n

      f)<\/strong> Inverse of 18 mod 32:<\/h3>\n\n\n\n

      $$
      \\gcd(18, 32) = 2 \\neq 1 \\Rightarrow \\text{No inverse.}
      $$<\/p>\n\n\n\n

      \n\n\n\n

      Explanation <\/strong><\/h3>\n\n\n\n

      The concept of a multiplicative inverse in modular arithmetic<\/strong> is foundational in number theory and cryptography. In the set $\\mathbb{Z}_n$, which contains integers modulo $n$, an element $a$ has a multiplicative inverse if there exists some integer $b$ such that $ab \\equiv 1 \\mod n$. This is only possible when $a$ and $n$ are coprime<\/strong>\u2014that is, $\\gcd(a, n) = 1$.<\/p>\n\n\n\n

      To find an inverse<\/strong>, you can either use trial and error (as in part b), or the Extended Euclidean Algorithm<\/strong>, also called the Pulverizer. The Pulverizer expresses $\\gcd(a, n) = 1$ as a linear combination $ax + ny = 1$, from which $x$ gives the inverse of $a$ modulo $n$.<\/p>\n\n\n\n

      In this set:<\/p>\n\n\n\n

        \n
      • Part b<\/strong> uses trial and error to find the inverse of 3 mod 16, then solves a linear congruence.<\/li>\n\n\n\n
      • Parts c<\/strong> and e<\/strong> involve using the Pulverizer to find the inverse and then solve equations.<\/li>\n\n\n\n
      • Parts d<\/strong> and f<\/strong> show cases where the inverse does not<\/strong> exist because the number and the modulus are not coprime.<\/li>\n<\/ul>\n\n\n\n

        Understanding when an inverse exists and how to compute it is crucial for solving modular equations and has real-world applications in cryptography (RSA encryption)<\/strong>, modular algebra<\/strong>, and computer science algorithms<\/strong>.<\/p>\n\n\n\n

        \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

        inverse in Zn? b) Use the definition of multiplicative inverse and trial and error to find the multiplicative inverse of 3 in Z16. Use the multiplicative inverse to solve 3x + 9 = 15 in Z16. c) Use the Pulverizer to find the multiplicative inverse of 37 in Z45 or explain why you cannot do […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-220203","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/220203","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=220203"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/220203\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=220203"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=220203"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=220203"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}