tag:blogger.com,1999:blog-63177295752340110462024-03-13T17:38:46.416+05:30gadre MathematicsGadre Chintamanihttp://www.blogger.com/profile/15647769573938152632noreply@blogger.comBlogger2125tag:blogger.com,1999:blog-6317729575234011046.post-2584914948483116322010-06-27T20:34:00.004+05:302010-06-27T20:40:34.754+05:30Simplified DIVISIBILITY TESTS for 9, 3, 4 and 8<p class="MsoNormal" align="center" style="text-align:center;mso-layout-grid-align: none;text-autospace:none"><b><u><span style="font-size:18.0pt;font-family:Arial; color:red;mso-bidi-language:ML">Simplified </span></u></b><b><u><span style="font-size:18.0pt;font-family:Arial;mso-bidi-language:ML">DIVISIBILITY TESTS for 9, 3, 4 and 8<o:p></o:p></span></u></b></p> <p class="MsoNormal" align="center" style="text-align:center;mso-layout-grid-align: none;text-autospace:none"><b><u><span style="font-family:Arial;mso-bidi-language: ML"><o:p><span style="text-decoration:none"> </span></o:p></span></u></b></p> <p class="MsoNormal" align="center" style="text-align:center;mso-layout-grid-align: none;text-autospace:none"><b><span style="font-family:Arial;mso-bidi-language: ML"><a href="mailto:gadrechintamani@gmail.com">gadrechintamani@gmail.com</a></span></b><span style="font-family:Arial;mso-bidi-language:ML"><o:p></o:p></span></p><p class="MsoNormal" align="center" style="text-align:center;mso-layout-grid-align: none;text-autospace:none"><b><span class="Apple-style-span" style="font-family: 'trebuchet ms', verdana, arial, sans-serif; font-weight: normal; font-size: 15px; color: rgb(51, 51, 51); line-height: 22px; ">http://gadremathematics.blogspot.com/</span></b></p><p class="MsoNormal" align="center" style="text-align:center;mso-layout-grid-align: none;text-autospace:none"><b><span style="font-family:Arial;mso-bidi-language:ML">Conventional Test for 9 <span style="color:blue">(or 3)</span> –</span></b><span style="font-family:Arial; mso-bidi-language:ML"> If sum of the digits of the number is divisible by 9 <span style="color:blue">(3)</span> then number is divisible by 9 <span style="color:blue">(3)</span>. The number 2754 is divisible by 9 <span style="color:blue">(3)</span> as sum of its digits = 2+7+5+4 = 18 is divisible by 9 <span style="color:blue">(3)</span>.</span></p> <p class="MsoNormal" style="mso-layout-grid-align:none;text-autospace:none"><span style="font-family:Arial;mso-bidi-language:ML"><o:p> </o:p></span></p> <p class="MsoNormal" style="mso-layout-grid-align:none;text-autospace:none"><b><span style="font-family:Arial;mso-bidi-language:ML">Here 2, 7, 5 and 4 are remainders when place value of the digits namely 2000, 700, 50 and 4 are divided by 9</span></b><span style="font-family:Arial;mso-bidi-language:ML">. When the sum of these remainders is divisible by 9 <span style="color:blue">(3)</span>, the number is divisible by 9 <span style="color:blue">(3)</span>. <o:p></o:p></span></p> <p class="MsoNormal" style="mso-layout-grid-align:none;text-autospace:none"><span style="font-family:Arial;mso-bidi-language:ML"><o:p> </o:p></span></p> <p class="MsoNormal" style="mso-layout-grid-align:none;text-autospace:none"><b><span style="font-family:Arial;color:red;mso-bidi-language:ML">In simplified test for 9, before making the sum of digits, first discard the remainders giving sum equal to 9 or multiple of 9 (“In test for 3” – sum equal to 3 or multiple of 3).<o:p></o:p></span></b></p> <p class="MsoNormal" style="mso-layout-grid-align:none;text-autospace:none"><span style="font-family:Arial;mso-bidi-language:ML">7247583 → [<s style="text-line-through: double">72</s>]<span style="color:red">(4)</span>{</span><!--[if supportFields]><span style="'font-family:Arial;mso-bidi-language:ML'"><span style="'mso-element:field-begin'"></span> eq \o\ac(</span><span style="'font-size:24.0pt;mso-bidi-font-size:12.0pt;">◇</span><span style="'font-family:Arial;mso-bidi-language:ML'">,7)</span><![endif]--><span style="font-family:Arial;mso-bidi-language:ML">7</span><!--[if supportFields]><span style="'font-family:Arial;mso-bidi-language:ML'"><span style="'mso-element:field-end'"></span></span><![endif]--><span style="font-family:Arial;mso-bidi-language:ML">}<span style="color:red">(5)</span>{</span><!--[if supportFields]><span style="'font-family:Arial;mso-bidi-language:ML'"><span style="'mso-element:field-begin'"></span> eq \o\ac(</span><span style="'font-size:32.0pt;mso-bidi-font-size:12.0pt;">□</span><span style="'font-family:Arial;mso-bidi-language:ML'">,83)</span><![endif]--><span style="font-family:Arial;mso-bidi-language:ML">83</span><!--[if supportFields]><span style="'font-family:Arial;mso-bidi-language:ML'"><span style="'mso-element:field-end'"></span></span><![endif]--><span style="font-family:Arial;mso-bidi-language:ML">} → Discarding sums 9,<span style="color:red">9</span>, and 18, remainder is 0. Therefore the number is divisible by 9 <span style="color:blue">(3)</span>. <o:p></o:p></span></p> <p class="MsoNormal" style="mso-layout-grid-align:none;text-autospace:none"><span style="font-family:Arial;mso-bidi-language:ML">72476583→[<s style="text-line-through: double">72</s>]<span style="color:red">(4)</span>{</span><!--[if supportFields]><span style="'font-family:Arial;mso-bidi-language:ML'"><span style="'mso-element:field-begin'"></span> eq \o\ac(</span><span style="'font-size:24.0pt;mso-bidi-font-size:12.0pt;">◇</span><span style="'font-family:Arial;mso-bidi-language:ML'">,7)</span><![endif]--><span style="font-family:Arial;mso-bidi-language:ML">7</span><!--[if supportFields]><span style="'font-family:Arial;mso-bidi-language:ML'"><span style="'mso-element:field-end'"></span></span><![endif]--><span style="font-family:Arial;mso-bidi-language:ML">}<b><span style="color:blue">6</span></b><span style="color:red">(5)</span>{</span><!--[if supportFields]><span style="'font-family:Arial;mso-bidi-language:ML'"><span style="'mso-element:field-begin'"></span> eq \o\ac(</span><span style="'font-size:32.0pt;mso-bidi-font-size:12.0pt;">□</span><span style="'font-family:Arial;mso-bidi-language:ML'">,83)</span><![endif]--><span style="font-family:Arial;mso-bidi-language:ML">83</span><!--[if supportFields]><span style="'font-family:Arial;mso-bidi-language:ML'"><span style="'mso-element:field-end'"></span></span><![endif]--><span style="font-family:Arial;mso-bidi-language:ML">}→ Discarding sums 9,<span style="color:red">9</span>, and 18 Remainder is <span style="color:blue">6</span>. <span style="mso-spacerun:yes"> </span>The number 72476583 is not divisible by 9. It will give remainder 6 when divided by 9. But the number is divisible by 3 as the remainder 6 is divisible by 3.<o:p></o:p></span></p> <p class="MsoNormal" style="mso-layout-grid-align:none;text-autospace:none"><span style="font-family:Arial;mso-bidi-language:ML"><o:p> </o:p></span></p> <p class="MsoNormal"><b><span style="font-family:Arial;mso-bidi-language:ML">Conventional Test for 4 –</span></b><span style="font-family:Arial;mso-bidi-language:ML">When the number formed by last two digits of a number is divisible by 4 then the number is divisible by 4. <o:p></o:p></span></p> <p class="MsoNormal"><span style="font-family:Arial;mso-bidi-language:ML"><o:p> </o:p></span></p> <p class="MsoNormal"><b><span style="font-family:Arial;color:red;mso-bidi-language: ML">In simplified approach this two digit number is written as sum of numbers.<o:p></o:p></span></b></p> <p class="MsoNormal"><span style="font-family:Arial;mso-bidi-language:ML">6936 is divisible by 4 as 36 is divisible by 4.<o:p></o:p></span></p> <p class="MsoNormal"><span style="font-family:Arial;mso-bidi-language:ML">2376 is divisible by 4 as 76 = 40 + 36 and both 40 and 36 are divisible by 4. <o:p></o:p></span></p> <p class="MsoNormal"><span style="font-family:Arial;mso-bidi-language:ML">8992 is divisible by 4 as 92 = 40 + 40 +12 and 40, 40and 12 are all divisible by 4. <o:p></o:p></span></p> <p class="MsoNormal"><b><span style="font-family:Arial;mso-bidi-language:ML"><o:p> </o:p></span></b></p> <p class="MsoNormal"><b><span style="font-family:Arial;mso-bidi-language:ML">Conventional Test for 8 –</span></b><span style="font-family:Arial;mso-bidi-language:ML">When the number formed by last three digits of a number is divisible by 8 then the number is divisible by 8. <o:p></o:p></span></p> <p class="MsoNormal"><span style="font-family:Arial;mso-bidi-language:ML"><o:p> </o:p></span></p> <p class="MsoNormal"><b><span style="font-family:Arial;color:red;mso-bidi-language: ML">In Simplified approach “last two digits” or “last two digits plus 4” are tested for 8 depending on whether digit at hundred is Even or Odd.<o:p></o:p></span></b></p> <p class="MsoNormal"><span style="font-family:Arial;mso-bidi-language:ML">(A<span style="color:blue">) <u>Even hundreds</u></span> are divisible by 8.Hence check only the number formed by <span style="color:blue">last two digits</span> is divisible by 8. <o:p></o:p></span></p> <p class="MsoNormal"><span style="font-family:Arial;mso-bidi-language:ML">(B) </span><u><span style="font-family:Arial;mso-bidi-font-family:"Times New Roman";color:red">Odd hundreds</span></u><span style="font-family:Arial;mso-bidi-font-family:"Times New Roman""> give remainder 4 when divided by 8. Hence check the sum of the number formed by <span style="color:red">last two digits and remainder 4</span> is divisible by 8.<o:p></o:p></span></p> <p class="MsoNormal"><span style="font-family:Arial;mso-bidi-font-family:"Times New Roman"">21</span><b><span style="font-size:14.0pt;font-family:Arial;mso-bidi-font-family:"Times New Roman"; color:blue">6</span></b><span style="font-family:Arial;mso-bidi-font-family: "Times New Roman"">56 </span><span style="font-family:Arial;mso-bidi-language: ML">→ </span><span style="font-family:Arial;mso-bidi-font-family:"Times New Roman"; color:blue">56</span><span style="font-family:Arial;mso-bidi-font-family:"Times New Roman""> and hence the number is divisible by 8.<o:p></o:p></span></p> <p class="MsoNormal"><span style="font-family:Arial;mso-bidi-font-family:"Times New Roman"">82</span><b><span style="font-size:14.0pt;font-family:Arial;mso-bidi-font-family:"Times New Roman"; color:blue">6</span></b><span style="font-family:Arial;mso-bidi-font-family: "Times New Roman"">86 </span><span style="font-family:Arial;mso-bidi-language: ML">→ <span style="color:blue">8</span></span><span style="font-family:Arial; mso-bidi-font-family:"Times New Roman";color:blue">6</span><span style="font-family:Arial;mso-bidi-font-family:"Times New Roman""> and hence the number is NOT divsible by 8.<o:p></o:p></span></p> <p class="MsoNormal"><span style="font-family:Arial;mso-bidi-font-family:"Times New Roman"">51</span><b><span style="font-size:14.0pt;font-family:Arial;mso-bidi-font-family:"Times New Roman"; color:red">7</span></b><span style="font-family:Arial;mso-bidi-font-family: "Times New Roman"">44 </span><span style="font-family:Arial;mso-bidi-language: ML">→ </span><span style="font-family:Arial;mso-bidi-font-family:"Times New Roman""><span style="mso-spacerun:yes"> </span><span style="color:red">44 + 4 = 48</span> and hence the number is divisible by 8. <o:p></o:p></span></p> <p class="MsoNormal"><span style="font-family:Arial;mso-bidi-font-family:"Times New Roman"">89</span><b><span style="font-size:14.0pt;font-family:Arial;mso-bidi-font-family:"Times New Roman"; color:red">3</span></b><span style="font-family:Arial;mso-bidi-font-family: "Times New Roman"">24 </span><span style="font-family:Arial;mso-bidi-language: ML">→ <span style="color:red">24</span></span><span style="font-family:Arial; mso-bidi-font-family:"Times New Roman";color:red"> + 4 = 28</span><span style="font-family:Arial;mso-bidi-font-family:"Times New Roman""> and hence the number is NOT divsible by 8.<o:p></o:p></span></p>Gadre Chintamanihttp://www.blogger.com/profile/15647769573938152632noreply@blogger.com0tag:blogger.com,1999:blog-6317729575234011046.post-81095105179303449592010-02-12T23:21:00.000+05:302010-02-12T23:23:10.302+05:30IntroHi,<div><br /></div><div>new to bolgging.</div><br /><br />Interested in Math PuzzlesGadre Chintamanihttp://www.blogger.com/profile/15647769573938152632noreply@blogger.com0