{"id":7344,"date":"2021-01-08T00:54:23","date_gmt":"2021-01-07T19:24:23","guid":{"rendered":"https:\/\/edu.janbal.org\/?p=7344"},"modified":"2021-01-08T00:54:23","modified_gmt":"2021-01-07T19:24:23","slug":"number-system","status":"publish","type":"post","link":"https:\/\/timesdarpan.com\/hi\/logical-mathematics\/number-system\/","title":{"rendered":"Aptitude &#8211; Number system"},"content":{"rendered":"\n<h2 class=\"has-vivid-cyan-blue-color has-text-color wp-block-heading\" style=\"font-size:22px\">Number system<\/h2>\n\n\n\n<p>In Decimal number system, there are ten symbols namely 0,1,2,3,4,5,6,7,8 and 9 called digits. A number is denoted by group of these digits called as numerals.<\/p>\n\n\n\n<h2 class=\"has-vivid-red-color has-text-color wp-block-heading\" style=\"font-size:18px\">Face Value<\/h2>\n\n\n\n<p>Face value of a digit in a numeral is value of the digit itself. For example in 321, face value of 1 is 1, face value of 2 is 2 and face value of 3 is 3.<\/p>\n\n\n\n<h2 class=\"has-vivid-red-color has-text-color wp-block-heading\" style=\"font-size:18px\">Place Value<\/h2>\n\n\n\n<p>Place value of a digit in a numeral is value of the digit multiplied by 10<sup>n<\/sup>&nbsp;where n starts from 0. For example in 321:<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>Place value of 1 = 1 x 10<sup>0<\/sup>&nbsp;= 1 x 1 = 1<\/li><li>Place value of 2 = 2 x 10<sup>1<\/sup>&nbsp;= 2 x 10 = 20<\/li><li>Place value of 3 = 3 x 10<sup>2<\/sup>&nbsp;= 3 x 100 = 300<\/li><\/ul>\n\n\n\n<p class=\"has-black-color has-text-color has-background\" style=\"background-color:#e8e9ea\">0<sup>th<\/sup>&nbsp;position digit is called unit digit and is the most commonly used topic in <a href=\"https:\/\/timesdarpan.com\/hi\/category\/aptitute\/\">aptitude<\/a> tests.<\/p>\n\n\n\n<h2 class=\"has-vivid-cyan-blue-color has-text-color has-medium-font-size wp-block-heading\">Types of Numbers<\/h2>\n\n\n\n<ol class=\"wp-block-list\"><li><strong>Natural Numbers<\/strong>&nbsp;&#8211; n &gt; 0 where n is counting number; [1,2,3&#8230;]<\/li><li><strong>Whole Numbers<\/strong>&nbsp;&#8211; n \u2265 0 where n is counting number; [0,1,2,3&#8230;].<\/li><\/ol>\n\n\n\n<p class=\"has-black-color has-text-color has-background\" style=\"background-color:#e8e9ea\">0 is the only whole number which is not a natural number. <br \/>Every natural number is a whole number.<\/p>\n\n\n\n<ol class=\"wp-block-list\" start=\"3\"><li><strong>Integers<\/strong>&nbsp;&#8211; n \u2265 0 or n \u2264 0 where n is counting number;&#8230;,-3,-2,-1,0,1,2,3&#8230; are integers.<\/li><\/ol>\n\n\n\n<ul class=\"wp-block-list\"><li><strong>Positive Integers<\/strong>&nbsp;&#8211; n &gt; 0; [1,2,3&#8230;]<\/li><li><strong>Negative Integers<\/strong>&nbsp;&#8211; n &lt; 0; [-1,-2,-3&#8230;]<\/li><li><strong>Non-Positive Integers<\/strong>&nbsp;&#8211; n \u2264 0; [0,-1,-2,-3&#8230;]<\/li><li><strong>Non-Negative Integers<\/strong>&nbsp;&#8211; n \u2265 0; [0,1,2,3&#8230;]   <span class=\"has-inline-color has-white-color\">number system<\/span><\/li><\/ul>\n\n\n\n<p class=\"has-black-color has-text-color has-background\" style=\"background-color:#e8e9ea\">0 is neither positive nor negative integer.<\/p>\n\n\n\n<ol class=\"wp-block-list\" start=\"4\"><li><strong>Even Numbers<\/strong>&nbsp;&#8211; n \/ 2 = 0 where n is counting number; [0,2,4,&#8230;]<\/li><li><strong>Odd Numbers<\/strong>&nbsp;&#8211; n \/ 2 \u2260 0 where n is counting number; [1,3,5,&#8230;]<\/li><li><strong>Prime Numbers<\/strong>&nbsp;&#8211; Numbers which is divisible by themselves only apart from 1.<\/li><\/ol>\n\n\n\n<p class=\"has-black-color has-text-color has-background\" style=\"background-color:#e8e9ea\">1 is not a prime number.<br \/><br \/>To test a number p to be prime, find a whole number k such that k &gt; \u221ap. Get all prime numbers less than or equal to k and divide p with each of these prime numbers. If no number divides p exactly then p is a prime number otherwise it is not a prime number.<\/p>\n\n\n\n<p class=\"has-black-color has-luminous-vivid-amber-background-color has-text-color has-background\">Example: 191 is prime number or not? <br \/>Solution:  <br \/>Step 1 &#8211; 14 &gt; \u221a191 <br \/>Step 2 &#8211; Prime numbers less than 14 are 2,3,5,7,11 and 13. <br \/>Step 3 &#8211; 191 is not divisible by any above prime number. <br \/>Result &#8211; 191 is a prime number. <br \/><br \/>Example: 187 is prime number or not? <br \/>Solution:  <br \/>Step 1 &#8211; 14 &gt; \u221a187 <br \/>Step 2 &#8211; Prime numbers less than 14 are 2,3,5,7,11 and 13. <br \/>Step 3 &#8211; 187 is divisible by 11. <br \/>Result &#8211; 187 is not a prime number.<\/p>\n\n\n\n<ol class=\"wp-block-list\" start=\"7\"><li><strong>Composite Numbers<\/strong>&nbsp;&#8211; Non-prime numbers &gt; 1. For example, 4,6,8,9 etc.<\/li><\/ol>\n\n\n\n<p class=\"has-background\" style=\"background-color:#e8e9ea\">1 is neither a prime number nor a composite number.<br \/>2 is the only even prime number.<\/p>\n\n\n\n<ol class=\"wp-block-list\" start=\"8\"><li><strong>Co-Primes Numbers<\/strong>&nbsp;&#8211; Two natural numbers are co-primes if their H.C.F. is 1. For example, (2,3), (4,5) are co-primes.<\/li><\/ol>\n\n\n\n<h2 class=\"has-vivid-cyan-blue-color has-text-color has-medium-font-size wp-block-heading\">Divisibility  <span class=\"has-inline-color has-white-color\">number system<\/span><\/h2>\n\n\n\n<p>Following are tips to check divisibility of numbers. <span class=\"has-inline-color has-white-color\">number system<\/span><\/p>\n\n\n\n<ol class=\"wp-block-list\"><li><strong>Divisibility by 2<\/strong>&nbsp;&#8211; A number is divisible by 2 if its unit digit is 0,2,4,6 or 8.<\/li><\/ol>\n\n\n\n<p class=\"has-black-color has-luminous-vivid-amber-background-color has-text-color has-background\">Example: 64578 is divisible by 2 or not?<br \/>Solution: <br \/>Step 1 &#8211; Unit digit is 8.<br \/>Result &#8211; 64578 is divisible by 2.<br \/><br \/>Example: 64575 is divisible by 2 or not?<br \/>Solution: <br \/>Step 1 &#8211; Unit digit is 5.<br \/>Result &#8211; 64575 is not divisible by 2.<\/p>\n\n\n\n<ol class=\"wp-block-list\" start=\"2\"><li><strong>Divisibility by 3<\/strong>&nbsp;&#8211; A number is divisible by 3 if sum of its digits is completely divisible by 3.<\/li><\/ol>\n\n\n\n<p class=\"has-black-color has-luminous-vivid-amber-background-color has-text-color has-background\">Example: 64578 is divisible by 3 or not?<br \/>Solution: <br \/>Step 1 &#8211; Sum of its digits is 6 + 4 + 5 + 7 + 8 = 30<br \/>which is divisible by 3.<br \/>Result &#8211; 64578 is divisible by 3.<br \/><br \/>Example: 64576 is divisible by 3 or not?<br \/>Solution: <br \/>Step 1 &#8211; Sum of its digits is 6 + 4 + 5 + 7 + 6 = 28<br \/>which is not divisible by 3.<br \/>Result &#8211; 64576 is not divisible by 3.<\/p>\n\n\n\n<ol class=\"wp-block-list\" start=\"3\"><li><strong>Divisibility by 4<\/strong>&nbsp;&#8211; A number is divisible by 4 if number formed using its last two digits is completely divisible by 4.<\/li><\/ol>\n\n\n\n<p class=\"has-black-color has-luminous-vivid-amber-background-color has-text-color has-background\">Example: 64578 is divisible by 4 or not?<br \/>Solution:<br \/>Step 1 &#8211; number formed using its last two digits is 78<br \/>which is not divisible by 4.<br \/>Result &#8211; 64578 is not divisible by 4.<br \/><br \/>Example: 64580 is divisible by 4 or not?<br \/>Solution:<br \/>Step 1 &#8211; number formed using its last two digits is 80<br \/>which is divisible by 4.<br \/>Result &#8211; 64580 is divisible by 4.<\/p>\n\n\n\n<ol class=\"wp-block-list\" start=\"4\"><li><strong>Divisibility by 5<\/strong>&nbsp;&#8211; A number is divisible by 5 if its unit digit is 0 or 5.<\/li><\/ol>\n\n\n\n<p class=\"has-black-color has-luminous-vivid-amber-background-color has-text-color has-background\">Example: 64578 is divisible by 5 or not?<br \/>Solution:<br \/>Step 1 &#8211; Unit digit is 8.<br \/>Result &#8211; 64578 is not divisible by 5.<br \/><br \/>Example: 64575 is divisible by 5 or not?<br \/>Solution: <br \/>Step 1 &#8211; Unit digit is 5.<br \/>Result &#8211; 64575 is divisible by 5.<\/p>\n\n\n\n<ol class=\"wp-block-list\" start=\"5\"><li><strong>Divisibility by 6<\/strong>&nbsp;&#8211; A number is divisible by 6 if the number is divisible by both 2 and 3. <span class=\"has-inline-color has-white-color\">number system<\/span><\/li><\/ol>\n\n\n\n<p class=\"has-black-color has-luminous-vivid-amber-background-color has-text-color has-background\">Example: 64578 is divisible by 6 or not?<br \/>Solution: <br \/>Step 1 &#8211; Unit digit is 8. Number is divisible by 2.<br \/>Step 2 &#8211; Sum of its digits is 6 + 4 + 5 + 7 + 8 = 30<br \/>which is divisible by 3.<br \/>Result &#8211; 64578 is divisible by 6.<br \/><br \/>Example: 64576 is divisible by 6 or not?<br \/>Solution: <br \/>Step 1 &#8211; Unit digit is 8. Number is divisible by 2.<br \/>Step 2 &#8211; Sum of its digits is 6 + 4 + 5 + 7 + 6 = 28 <br \/>which is not divisible by 3.<br \/>Result &#8211; 64576 is not divisible by 6.<\/p>\n\n\n\n<ol class=\"wp-block-list\" start=\"6\"><li><strong>Divisibility by 8<\/strong>&nbsp;&#8211; A number is divisible by 8 if number formed using its last three digits is completely divisible by 8.<\/li><\/ol>\n\n\n\n<p class=\"has-black-color has-luminous-vivid-amber-background-color has-text-color has-background\">Example: 64578 is divisible by 8 or not?<br \/>Solution:<br \/>Step 1 &#8211; number formed using its last three digits is 578<br \/>which is not divisible by 8.<br \/>Result &#8211; 64578 is not divisible by 8.<br \/><br \/>Example: 64576 is divisible by 8 or not?<br \/>Solution: <br \/>Step 1 &#8211; number formed using its last three digits is 576<br \/>which is divisible by 8.<br \/>Result &#8211; 64576 is divisible by 8.<\/p>\n\n\n\n<ol class=\"wp-block-list\" start=\"7\"><li><strong>Divisibility by 9<\/strong>&nbsp;&#8211; A number is divisible by 9 if sum of its digits is completely divisible by 9.<\/li><\/ol>\n\n\n\n<p class=\"has-black-color has-luminous-vivid-amber-background-color has-text-color has-background\">Example: 64579 is divisible by 9 or not?<br \/>Solution: <br \/>Step 1 &#8211; Sum of its digits is 6 + 4 + 5 + 7 + 9 = 31 <br \/>which is not divisible by 9.<br \/>Result &#8211; 64579 is not divisible by 9.<br \/><br \/>Example: 64575 is divisible by 9 or not?<br \/>Solution: <br \/>Step 1 &#8211; Sum of its digits is 6 + 4 + 5 + 7 + 5 = 27<br \/>which is divisible by 9.<br \/>Result &#8211; 64575 is divisible by 9.<\/p>\n\n\n\n<ol class=\"wp-block-list\" start=\"8\"><li><strong>Divisibility by 10<\/strong>&nbsp;&#8211; A number is divisible by 10 if its unit digit is 0.<\/li><\/ol>\n\n\n\n<p class=\"has-black-color has-luminous-vivid-amber-background-color has-text-color has-background\">Example: 64575 is divisible by 10 or not?<br \/>Solution:<br \/>Step 1 &#8211; Unit digit is 5.<br \/>Result &#8211; 64578 is not divisible by 10.<br \/><br \/>Example: 64570 is divisible by 10 or not?<br \/>Solution: <br \/>Step 1 &#8211; Unit digit is 0.<br \/>Result &#8211; 64570 is divisible by 10.<\/p>\n\n\n\n<ol class=\"wp-block-list\" start=\"9\"><li><strong>Divisibility by 11<\/strong>&nbsp;&#8211; A number is divisible by 11 if difference between sum of digits at odd places and sum of digits at even places is either 0 or is divisible by 11.<\/li><\/ol>\n\n\n\n<p class=\"has-black-color has-luminous-vivid-amber-background-color has-text-color has-background\">Example: 64575 is divisible by 11 or not?<br \/>Solution: <br \/>Step 1 &#8211; difference between sum of digits at odd places <br \/> and sum of digits at even places = (6+5+5) &#8211; (4+7) = 5 <br \/>which is not divisible by 11.<br \/>Result &#8211; 64575 is not divisible by 11.<br \/><br \/>Example: 64075 is divisible by 11 or not?<br \/>Solution: <br \/>Step 1 &#8211; difference between sum of digits at odd places <br \/> and sum of digits at even places = (6+0+5) &#8211; (4+7) = 0.<br \/>Result &#8211; 64075 is divisible by 11.<\/p>\n\n\n\n<h2 class=\"has-vivid-cyan-blue-color has-text-color has-medium-font-size wp-block-heading\">Tips on Division <span class=\"has-inline-color has-white-color\">number system<\/span><\/h2>\n\n\n\n<ol class=\"wp-block-list\"><li>If a number n is divisible by two co-primes numbers a, b then n is divisible by ab.<\/li><li>(a-b) always divides (a<sup>n<\/sup>&nbsp;&#8211; b<sup>n<\/sup>) if n is a natural number.<\/li><li>(a+b) always divides (a<sup>n<\/sup>&nbsp;&#8211; b<sup>n<\/sup>) if n is an even number.<\/li><li>(a+b) always divides (a<sup>n<\/sup>&nbsp;+ b<sup>n<\/sup>) if n is an odd number. <span class=\"has-inline-color has-white-color\">number system<\/span><\/li><\/ol>\n\n\n\n<h2 class=\"has-vivid-cyan-blue-color has-text-color wp-block-heading\" style=\"font-size:19px\">Division Algorithm<\/h2>\n\n\n\n<p>When a number is divided by another number then<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>Dividend = (Divisor x Quotient) + Reminder<\/strong><\/p>\n\n\n\n<h2 class=\"has-vivid-cyan-blue-color has-text-color wp-block-heading\" style=\"font-size:19px\">Series<\/h2>\n\n\n\n<p>Following are formulaes for basic number series:<\/p>\n\n\n\n<ol class=\"wp-block-list\"><li>(1+2+3+&#8230;+n) = (1\/2)n(n+1)<\/li><li>(1<sup>2<\/sup>+2<sup>2<\/sup>+3<sup>2<\/sup>+&#8230;+n<sup>2<\/sup>) = (1\/6)n(n+1)(2n+1)<\/li><li>(1<sup>3<\/sup>+2<sup>3<\/sup>+3<sup>3<\/sup>+&#8230;+n<sup>3<\/sup>) = (1\/4)n<sup>2<\/sup>(n+1)<sup>2<\/sup><\/li><\/ol>\n\n\n\n<h2 class=\"has-vivid-cyan-blue-color has-text-color wp-block-heading\" style=\"font-size:19px\">Basic Formulas<\/h2>\n\n\n\n<p>These are the <a href=\"https:\/\/www.facebook.com\/Times-Darpan-104439664934487\" target=\"_blank\" rel=\"noopener\">basic<\/a> formulae:<\/p>\n\n\n\n<p class=\"has-black-color has-text-color has-background\" style=\"background-color:#e8e9ea\"><strong>(a + b)<sup>2<\/sup> = a<sup>2<\/sup> + b<sup>2<\/sup> + 2ab<br \/><br \/>(a &#8211; b)<sup>2<\/sup> = a<sup>2<\/sup> + b<sup>2<\/sup> &#8211; 2ab<br \/><br \/>(a + b)<sup>2<\/sup> &#8211; (a &#8211; b)<sup>2<\/sup> = 4ab<br \/><br \/>(a + b)<sup>2<\/sup> + (a &#8211; b)<sup>2<\/sup> = 2(a<sup>2<\/sup> + b<sup>2<\/sup>)<br \/><br \/>(a<sup>2<\/sup> &#8211; b<sup>2<\/sup>) = (a + b)(a &#8211; b)<br \/><br \/>(a + b + c)<sup>2<\/sup> = a<sup>2<\/sup> + b<sup>2<\/sup> + c<sup>2<\/sup> + 2(ab + bc + ca)<br \/><br \/>(a<sup>3<\/sup> + b<sup>3<\/sup>) = (a + b)(a<sup>2<\/sup> &#8211;  ab + b<sup>2<\/sup>)<br \/><br \/>(a<sup>3<\/sup> &#8211; b<sup>3<\/sup>) = (a &#8211; b)(a<sup>2<\/sup> + ab + b<sup>2<\/sup>)<br \/><br \/>(a<sup>3<\/sup> + b<sup>3<\/sup> + c<sup>3<\/sup> &#8211; 3abc) = (a + b + c)(a<sup>2<\/sup> + b<sup>2<\/sup> + c<sup>2<\/sup> &#8211; ab &#8211; bc &#8211; ca)<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Number system In Decimal number system, there are ten symbols namely 0,1,2,3,4,5,6,7,8 and 9 called digits. A number is denoted by group of these digits called as numerals. Face Value Face value of a digit in a numeral is value of the digit itself. For example in 321, face value of 1 is 1, face [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":7347,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"fifu_image_url":"","fifu_image_alt":"","footnotes":""},"categories":[390],"tags":[583],"class_list":{"0":"post-7344","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-logical-mathematics","8":"tag-number-system"},"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/timesdarpan.com\/hi\/wp-json\/wp\/v2\/posts\/7344","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/timesdarpan.com\/hi\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/timesdarpan.com\/hi\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/timesdarpan.com\/hi\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/timesdarpan.com\/hi\/wp-json\/wp\/v2\/comments?post=7344"}],"version-history":[{"count":0,"href":"https:\/\/timesdarpan.com\/hi\/wp-json\/wp\/v2\/posts\/7344\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/timesdarpan.com\/hi\/wp-json\/"}],"wp:attachment":[{"href":"https:\/\/timesdarpan.com\/hi\/wp-json\/wp\/v2\/media?parent=7344"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/timesdarpan.com\/hi\/wp-json\/wp\/v2\/categories?post=7344"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/timesdarpan.com\/hi\/wp-json\/wp\/v2\/tags?post=7344"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}