To solve the congruence 3x\u22617(mod11)3x \\equiv 7 \\pmod{11}3x\u22617(mod11), we aim to find an integer xxx that satisfies this congruence. Here are the steps to solve it:<\/p>\n\n\n\n
Step 1: Find the inverse of 3 modulo 11<\/h3>\n\n\n\n
The first step is to find the modular inverse of 3 modulo 11. The modular inverse of 3 is a number yyy such that:3y\u22611(mod11)3y \\equiv 1 \\pmod{11}3y\u22611(mod11)<\/p>\n\n\n\n
To find yyy, we can use the Extended Euclidean Algorithm. We need to find integers yyy and kkk such that:3y\u221211k=13y – 11k = 13y\u221211k=1<\/p>\n\n\n\n
Perform the Euclidean algorithm to find the greatest common divisor (gcd) of 3 and 11:11=3\u00d73+2(divide 11 by 3)11 = 3 \\times 3 + 2 \\quad \\text{(divide 11 by 3)}11=3\u00d73+2(divide 11 by 3)3=2\u00d71+1(divide 3 by 2)3 = 2 \\times 1 + 1 \\quad \\text{(divide 3 by 2)}3=2\u00d71+1(divide 3 by 2)2=1\u00d72+0(divide 2 by 1, remainder 0, so gcd is 1)2 = 1 \\times 2 + 0 \\quad \\text{(divide 2 by 1, remainder 0, so gcd is 1)}2=1\u00d72+0(divide 2 by 1, remainder 0, so gcd is 1)<\/p>\n\n\n\n
Now, work backwards to express 1 as a linear combination of 3 and 11:1=3\u22121\u00d721 = 3 – 1 \\times 21=3\u22121\u00d72<\/p>\n\n\n\n
Substitute 2=11\u22123\u00d732 = 11 – 3 \\times 32=11\u22123\u00d73:1=3\u22121\u00d7(11\u22123\u00d73)1 = 3 – 1 \\times (11 – 3 \\times 3)1=3\u22121\u00d7(11\u22123\u00d73)1=3\u22121\u00d711+3\u00d731 = 3 – 1 \\times 11 + 3 \\times 31=3\u22121\u00d711+3\u00d731=4\u00d73\u22121\u00d7111 = 4 \\times 3 – 1 \\times 111=4\u00d73\u22121\u00d711<\/p>\n\n\n\n
Thus, the inverse of 3 modulo 11 is 444, because 4\u00d73\u22611(mod11)4 \\times 3 \\equiv 1 \\pmod{11}4\u00d73\u22611(mod11).<\/p>\n\n\n\n
Step 2: Multiply both sides of the congruence by 4<\/h3>\n\n\n\n
Now that we know the inverse of 3 modulo 11 is 4, we multiply both sides of the congruence 3x\u22617(mod11)3x \\equiv 7 \\pmod{11}3x\u22617(mod11) by 4:4\u00d73x\u22614\u00d77(mod11)4 \\times 3x \\equiv 4 \\times 7 \\pmod{11}4\u00d73x\u22614\u00d77(mod11)12x\u226128(mod11)12x \\equiv 28 \\pmod{11}12x\u226128(mod11)<\/p>\n\n\n\n
Since 12\u22611(mod11)12 \\equiv 1 \\pmod{11}12\u22611(mod11), we simplify the left-hand side:x\u226128(mod11)x \\equiv 28 \\pmod{11}x\u226128(mod11)<\/p>\n\n\n\n
Now, reduce 28 modulo 11:28\u00f711=2 remainder 628 \\div 11 = 2 \\text{ remainder } 628\u00f711=2 remainder 6<\/p>\n\n\n\n
So, 28\u22616(mod11)28 \\equiv 6 \\pmod{11}28\u22616(mod11).<\/p>\n\n\n\n
Thus, the solution to the congruence is:x\u22616(mod11)x \\equiv 6 \\pmod{11}x\u22616(mod11)<\/p>\n\n\n\n
Final Answer:<\/h3>\n\n\n\n
The solution to 3x\u22617(mod11)3x \\equiv 7 \\pmod{11}3x\u22617(mod11) is x\u22616(mod11)x \\equiv 6 \\pmod{11}x\u22616(mod11). Therefore, x=6x = 6x=6 is the smallest non-negative solution, and any integer xxx of the form x=6+11kx = 6 + 11kx=6+11k, where kkk is any integer, is also a solution.<\/p>\n\n\n\n Find the solution of the congruence 3x \u2261 7 (mod 11) The Correct Answer and Explanation is: To solve the congruence 3x\u22617(mod11)3x \\equiv 7 \\pmod{11}3x\u22617(mod11), we aim to find an integer xxx that satisfies this congruence. Here are the steps to solve it: Step 1: Find the inverse of 3 modulo 11 The first step […]<\/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-245872","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/245872","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=245872"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/245872\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=245872"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=245872"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=245872"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}![\"\"](\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/07\/learnexams-banner5-736.jpeg\")
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