Book Store

Download books and chapters from book store.
Currently only available for
CBSE Gujarat Board Haryana Board

Previous Year Papers

Download the PDF Question Papers Free for off line practice and view the Solutions online.
Currently only available for
Class 10 Class 12

યુક્લિડની ભાગવિધિ અને વાસ્તવિક સંખ્યાઓ

સાબિત કરો કે, એ n⁴ + 4n² +11 એ 16 વડે વિભાજ્ય છે, જ્યાં n એ અયુગ્મ પૂર્ણાંક છે.

અહીં n અયુગ્મ પૂર્ણાંક હોવાથી, કોઈક k $bold element of$ Z માટે,

n = 4k + 1 અથવા n = 4k – 1

વિકલ્પ 1 :

n = 4k + 1 માટે,

$bold n to the power of bold 4 bold space bold plus bold space bold 4 bold n to the power of bold 2 bold space bold plus bold space bold 11 bold equals bold space bold left parenthesis bold 4 bold k bold space bold space bold plus bold space bold 1 bold right parenthesis to the power of bold 4 bold space bold plus bold space bold 4 bold space bold space bold left parenthesis bold 4 bold k bold space bold plus bold space bold 1 bold right parenthesis to the power of bold 2 bold space bold plus bold space bold 11 bold equals bold space bold left parenthesis bold 4 bold k bold right parenthesis to the power of bold 4 bold space bold plus bold space bold 4 bold space bold left parenthesis bold 4 bold k bold right parenthesis to the power of bold 3 bold space bold plus bold space bold space bold 6 bold space bold left parenthesis bold 4 bold k bold right parenthesis to the power of bold 2 bold space bold plus bold space bold 4 bold space bold left parenthesis bold 4 bold k bold right parenthesis bold space bold plus bold space bold 1 bold space bold plus bold space bold 4 bold space bold left parenthesis bold 16 bold k to the power of bold 2 bold space bold plus bold space bold 8 bold k bold space bold plus bold space bold space bold 1 bold right parenthesis bold space bold plus bold space bold 11 bold left parenthesis bold space bold because bold space bold left parenthesis bold a bold space bold plus bold space bold b bold right parenthesis to the power of bold 4 bold space bold equals bold space bold a to the power of bold 4 bold space bold plus bold space bold space bold 4 bold a to the power of bold 3 bold b bold space bold plus bold space bold 6 bold a to the power of bold 2 bold b to the power of bold 2 bold space bold plus bold space bold 4 bold ab to the power of bold 3 bold space bold plus bold space bold b to the power of bold 4 bold right parenthesis bold equals bold space bold 256 bold k to the power of bold 4 bold space bold plus bold space bold 256 bold k to the power of bold 3 bold space bold plus bold space bold 96 bold k to the power of bold 2 bold space bold plus bold space bold 16 bold k bold space bold plus bold space bold 1 bold space bold plus bold space bold 64 bold k to the power of bold 2 bold space bold plus bold space bold 32 bold k bold space bold plus bold space bold 4 bold space bold plus bold space bold 11 bold equals bold space bold 256 bold k to the power of bold 4 bold space bold plus bold space bold 256 bold k to the power of bold 3 bold space bold plus bold space bold 160 bold k to the power of bold 2 bold space bold plus bold space bold 48 bold k bold space bold plus bold space bold 16 bold equals bold 16 bold space bold left parenthesis bold 16 bold k to the power of bold 4 bold space bold plus bold space bold 16 bold k to the power of bold 3 bold space bold plus bold space bold 10 bold k to the power of bold 2 bold space bold plus bold space bold 3 bold k bold space bold plus bold space bold 1 bold right parenthesis bold comma$

જે 16 વડે વિભાજ્ય છે.

વિકલ્પ 2 :

n = 4k - 1 માટે,

$bold n to the power of bold 4 bold space bold plus bold space bold 4 bold n to the power of bold 2 bold space bold plus bold space bold 11 bold equals bold space bold left parenthesis bold 4 bold k bold space bold minus bold space bold 1 bold right parenthesis to the power of bold 4 bold space bold plus bold space bold 4 bold space bold left parenthesis bold 4 bold k bold space bold minus bold space bold 1 bold right parenthesis to the power of bold 2 bold space bold plus bold space bold 11 bold equals bold space bold left parenthesis bold 4 bold k bold right parenthesis to the power of bold 4 bold space bold minus bold space bold 4 bold space bold left parenthesis bold 4 bold k bold right parenthesis to the power of bold 3 bold space bold plus bold space bold 6 bold space bold left parenthesis bold 4 bold k bold right parenthesis to the power of bold 2 bold space bold minus bold space bold 4 bold space bold left parenthesis bold 4 bold k bold right parenthesis bold space bold plus bold space bold 1 bold space bold plus bold space bold 4 bold space bold left parenthesis bold 16 bold k to the power of bold 2 bold space bold minus bold space bold 8 bold k bold space bold plus bold space bold 1 bold right parenthesis bold space bold plus bold space bold 11 bold left parenthesis bold because bold space bold left parenthesis bold a bold space bold minus bold space bold b bold right parenthesis to the power of bold 4 bold space bold equals bold space bold a to the power of bold 4 bold space bold minus bold space bold 4 bold a to the power of bold 3 bold b bold space bold plus bold space bold 6 bold a to the power of bold 2 bold b to the power of bold 2 bold space bold minus bold space bold 4 bold ab to the power of bold 3 bold space bold plus bold space bold b to the power of bold 4 bold equals bold space bold 256 bold k to the power of bold 4 bold space bold minus bold space bold 256 bold k to the power of bold 3 bold space bold plus bold space bold 96 bold k to the power of bold 2 bold space bold minus bold space bold 16 bold k bold space bold plus bold space bold 1 bold space bold plus bold space bold 64 bold k to the power of bold 2 bold space bold minus bold space bold 32 bold k bold space bold plus bold space bold 4 bold space bold plus bold space bold 11 bold equals bold space bold 256 bold k to the power of bold 4 bold space bold minus bold space bold 256 bold k to the power of bold 3 bold space bold plus bold space bold 160 bold k to the power of bold 2 bold space bold minus bold space bold 48 bold k bold space bold plus bold space bold 16 bold equals bold space bold 16 bold space bold left parenthesis bold 16 bold k to the power of bold 4 bold space bold minus bold space bold 16 bold k to the power of bold 3 bold space bold plus bold space bold 10 bold k to the power of bold 2 bold space bold minus bold space bold 3 bold k bold space bold plus bold space bold 1 bold right parenthesis bold comma bold space$
જે 16 વડે વિભાજ્ય છે.

આમ, કોઈ પણ વિકલ્પમાં n⁴ + 4n² + 11 એ 16 વડે વિભાજ્ય છે.

સાબિત કરો કે, 2a + 3b અને 9a + 5b માથી કોઈ એક 17 વડે વિભાજ્ય હોય તો બીજી અભિવ્યક્તિ પણ 17 વડે વિભાજ્ય છે; a, b $bold element of$ N.
(સુચન 4(2a + 3b) + 9a + 5b = 17a + 17b)