{"id":4289,"date":"2023-02-09T10:01:00","date_gmt":"2023-02-09T04:31:00","guid":{"rendered":"https:\/\/www.aplustopper.com\/?p=4289"},"modified":"2023-02-09T18:08:16","modified_gmt":"2023-02-09T12:38:16","slug":"finding-factors-multiples","status":"publish","type":"post","link":"https:\/\/www.aplustopper.com\/finding-factors-multiples\/","title":{"rendered":"Finding Factors And Multiples"},"content":{"rendered":"<h2><strong>FACTORS AND MULTIPLES<\/strong><\/h2>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-128301\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/04\/Finding-Factors-And-Multiples-1.jpg\" alt=\"Finding Factors And Multiples 1\" width=\"719\" height=\"430\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/04\/Finding-Factors-And-Multiples-1.jpg 719w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/04\/Finding-Factors-And-Multiples-1-300x179.jpg 300w\" sizes=\"(max-width: 719px) 100vw, 719px\" \/><br \/>\nMr Sharma wanted to withdraw Rs. 1000 from his\u00a0bank account to purchase books for his children.\u00a0The cashier gave him 10 hundred-rupee notes, i.e.,<br \/>\nRs. 10 \u00d7 100 = Rs. 1000<br \/>\nMr Sharma got the required amount. But the cashier could also give the same amount in the following ways:<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-128303\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/04\/Finding-Factors-And-Multiples-2.jpg\" alt=\"Finding Factors And Multiples 2\" width=\"245\" height=\"563\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/04\/Finding-Factors-And-Multiples-2.jpg 245w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/04\/Finding-Factors-And-Multiples-2-131x300.jpg 131w\" sizes=\"(max-width: 245px) 100vw, 245px\" \/><br \/>\nHere, we observe that in each case Mr Sharma got the same amount of Rs. 1000. These numbers 1, 2, 5, 10, 20, 50, 100, 200, 500, and 1000 are factors of 1000. Hence, 1000 is a multiple of these numbers.<br \/>\nHere we will discuss only the natural numbers, that is positive integers.<br \/>\nIf a = b \u00d7 c, we say b and c are factors of a and a is a multiple of c and b.<\/p>\n<p><iframe loading=\"lazy\" title=\"What are Multiples of a Number? How are Factors and Multiples related? | Don&#039;t Memorise\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/rUrLuTMq-sw?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen><\/iframe><\/p>\n<h2><strong>Factors<\/strong><\/h2>\n<p><strong>Factor:<\/strong> A number which divides a given number exactly (without leaving any remainder) is called a factor of the given number.<br \/>\n<strong>Example:<\/strong> Factors of 12<br \/>\n12 = 1 \u00d7 12<br \/>\n12 = 2 \u00d7 6<br \/>\n12 = 3 \u00d7 4<br \/>\nHere, 1, 2, 3, 4, 6, and 12 are factors of 12.<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/c1.staticflickr.com\/3\/2817\/34123575552_efb0a48cbe_o.png\" alt=\"finding-factors-multiples-2\" width=\"1066\" height=\"800\" \/><\/p>\n<h3><strong>Properties of Factors<\/strong><\/h3>\n<ol>\n<li>Every non-zero number is a factor of itself.<br \/>\n<strong>Examples:<\/strong> 5 is a factor of 5. (5 \u00f7 5 = 1)<br \/>\n12 is a factor of 12. (12 \u00f7 12 = 1)<\/li>\n<li>1 is a factor of every number.<br \/>\n<strong>Examples:<\/strong> 1 is a factor of 5. (5 \u00f7 1 = 5)<br \/>\n1 is a factor of 12. (12 \u00f7 1 = 12)<\/li>\n<li>Every non-zero number is a factor of 0.<br \/>\n<strong>Example:<\/strong> 5 and 12 are factors of 0 because<br \/>\n0 \u00f7 \u00a05 = 0 and 0 \u00f7 12 = 0<\/li>\n<li>The factors of a number are finite.<\/li>\n<\/ol>\n<h2><strong>Multiples<\/strong><\/h2>\n<p><strong>Multiple:<\/strong> A multiple of any natural number is a number formed by multiplying it by another natural number.<br \/>\n<strong>Example:<\/strong> Multiples of 6 are 6 \u00d7 1 = 6; 6 \u00d7 2 = 12; 6 \u00d7 3 = 18; 6 \u00d7 4 = 24<br \/>\nHere, 6,12,18,24 are multiples of 6.<br \/>\n<strong>Example:<\/strong> Let us find the LCM and HCF of 24 and 36.<br \/>\nFactors of 24 = 1, 2, 3, 4, 6, 8, 12, 24<br \/>\nFactors of 36 = 1, 2, 3, 4, 6, 9, 12, 18, 36<br \/>\nHere, the highest common factor is 12.<br \/>\n\u2234 HCF = 12<br \/>\nMultiples of 24 = 24, 48, 72, 96,&#8230;<br \/>\nMultiples of 36 = 36, 72, 108,&#8230;<br \/>\nHere, the lowest common multiple is 72.<br \/>\n\u2234 LCM = 72<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/c1.staticflickr.com\/3\/2861\/34280757755_8e738dedd2_o.png\" alt=\"finding-factors-multiples-3\" width=\"1071\" height=\"800\" \/><\/p>\n<h3><strong>Properties of Multiples<\/strong><\/h3>\n<ol>\n<li>Every number is a multiple of itself.<br \/>\n<strong>Examples<\/strong><br \/>\n(a) 3 \u00d7 1 = 3; 3 is the multiple of 3<br \/>\n(b) 7 \u00d7 1 = 7; 7 is the multiple of 7<\/li>\n<li>Every number is the multiple of 1.<br \/>\n<strong>Examples<\/strong><br \/>\n(a) 1 \u00d7 3 = 3; 3 is the multiple of 1<br \/>\n(b) 1 \u00d7 7 = 7; 7 is the multiple of 1<\/li>\n<li>The multiples of a number are infinite (unlimited).<\/li>\n<\/ol>\n<p><strong>Even numbers:<\/strong> A number which is a multiple of 2 is called an even number.<br \/>\n<strong>Example:<\/strong> 2, 4, 6, 8, 10,&#8230;<\/p>\n<p><strong>Odd numbers:<\/strong> A number which is not a multiple of 2 is called an odd number.<br \/>\n<strong>Example:<\/strong> 1, 3, 5, 7, 9, 11,&#8230;<\/p>\n<p><strong>Prime numbers:<\/strong> A number which is greater than 1, and has exactly two factors (1 and the number itself) is called a prime number.<br \/>\n<strong>Example:<\/strong> Factors of 2 = 1, 2<br \/>\nFactors of 3 = 1, 3<br \/>\nFactors of 5 = 1, 5<br \/>\nFactors of 7 = 1, 7<br \/>\nFactors of 11 = 1, 11<br \/>\nHere, 2, 3, 5, 7, 11 etc. are all prime numbers.<\/p>\n<p><strong>Composite numbers:<\/strong> A number, which is greater than 1 and has more than two factors is called a composite number.<br \/>\n<strong>Examples:<\/strong> Here,<br \/>\nFactors of 4 = 1, 2, 4<br \/>\nFactors of 6 = 1, 2, 3, 6<br \/>\nFactors of 8 = 1, 2, 4, 8<br \/>\nFactors of 9 = 1, 3, 9<br \/>\nFactors of 10 = 1, 2, 5<\/p>\n<p><strong>FINDING PRIME NUMBERS FROM 1 TO 100<\/strong><br \/>\nWe can find the prime numbers from 1 to 100 by following these steps (given by the Greek mathematician Eratosthenes).<br \/>\n<strong>Step 1:<\/strong> Prepare a list of numbers from 1 to 100.<br \/>\n<strong>Step 2:<\/strong> As 1 is neither prime nor composite number, cross it out.<br \/>\n<strong>Step 3:<\/strong> Encircle \u20182\u2019 as a prime number and cross out all its other multiples.<br \/>\n<strong>Step 4:<\/strong> Encircle \u20183\u2019 as a prime number and cross out all its other multiples.<br \/>\n<strong>Step 5:<\/strong> Encircle \u20185\u2019 as a prime number and cross out all its other multiples.<br \/>\n<strong>Step 6:<\/strong>\u00a0Continue this process till all the numbers are either encircled or crossed out.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-128304\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/04\/Finding-Factors-And-Multiples-3.png\" alt=\"Finding Factors And Multiples 3\" width=\"356\" height=\"311\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/04\/Finding-Factors-And-Multiples-3.png 356w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/04\/Finding-Factors-And-Multiples-3-300x262.png 300w\" sizes=\"(max-width: 356px) 100vw, 356px\" \/><br \/>\nAll the encircled numbers are <strong>prime numbers<\/strong> and the crossed out numbers (except 1) are <strong>composite numbers<\/strong>.<br \/>\nNumbers 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 are the prime numbers between 1 and 100.<br \/>\nThis is called the <strong>\u2018Sieve of Eratosthenes\u2019<\/strong>.<\/p>\n<p><strong>Twin primes:<\/strong> Two prime numbers having a difference of 2 are known as twin primes.<br \/>\n<strong>Example:<\/strong> ( 3, 5 ), ( 5,7 ), (11,13 ), (17,19 ), etc are twin primes.<\/p>\n<p><strong>Co-primes:<\/strong> Two numbers are said to be co-primes if they have no common factor other than 1. In other words, two natural numbers are co-primes if their HCF is 1.<br \/>\n<strong>Example:<\/strong> ( 2, 3 ), ( 3, 4 ), ( 5, 6 ), ( 7, 8 ), and so on.<\/p>\n<p><strong>Example 1:<\/strong> Is 16380 a multiple of 28?<br \/>\n<strong>Solution:<\/strong> To check whether 16380 is a multiple of 28 or not, we have to divide 16380 by 28. If the remainder becomes zero, then it is a multiple of the number.<br \/>\nSo, 16380 = 28 \u00d7\u00a0585, hence 16380 is a multiple of 28.<\/p>\n<p><strong>Example 2:<\/strong> Express 29 as the sum of three odd prime numbers.<br \/>\n<strong>Solution:\u00a0<\/strong>29 = 19 + 7 + 3<br \/>\nAll 19, 7, and 3 are odd prime numbers.<\/p>\n<h3><strong>DIVISIBILITY TESTS FOR 2, 3, 4, 5, 6, 7, 8, 9, 10, AND 11<\/strong><\/h3>\n<p>If we want to know that a number is divisible by another number, we generally perform the actual division and see whether the remainder is zero or not. This process is time-consuming for division of large numbers. Therefore, to cut short our efforts, some divisibility tests of different numbers are given below.<\/p>\n<table style=\"height: 1180px;\" border=\"2\" width=\"718\">\n<tbody>\n<tr>\n<td width=\"97\">\n<p style=\"text-align: center;\"><strong>Test of Divisibility by<\/strong><\/p>\n<\/td>\n<td style=\"text-align: center;\" width=\"272\"><strong>Condition<\/strong><\/td>\n<td width=\"302\">\n<p style=\"text-align: center;\"><strong>Example<\/strong><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td width=\"97\">\n<p style=\"text-align: center;\">2<\/p>\n<\/td>\n<td width=\"272\">A number is divisible by 2, if its ones digit is 0, 2, 4, 6 or 8.<\/td>\n<td width=\"302\">1372, 468, 500, 966 are divisible by 2, since their ones digit is 2, 8, 0 and 6 respectively.<\/td>\n<\/tr>\n<tr>\n<td width=\"97\">\n<p style=\"text-align: center;\">3<\/p>\n<\/td>\n<td width=\"272\">A number is divisible by 3, if the sum of its digits is divisible by 3.<\/td>\n<td width=\"302\">In 1881, the sum of digits is 1 + 8 + 8 + 1 = 18 which is divisible by 3. So 1881 is divisible by 3.<\/td>\n<\/tr>\n<tr>\n<td width=\"97\">\n<p style=\"text-align: center;\">4<\/p>\n<\/td>\n<td width=\"272\">A number is divisible by 4, if the number formed by the last two digits is divisible by 4.<\/td>\n<td width=\"302\">30776, 63784, 864 are all divisible by 4. Since last two digits of the numbers, i.e., 76, 84, and 64 are divisible by 4.<\/td>\n<\/tr>\n<tr>\n<td width=\"97\">\n<p style=\"text-align: center;\">5<\/p>\n<\/td>\n<td width=\"272\">A number is divisible by 5, if its ones digit is either 5 or 0.<\/td>\n<td width=\"302\">675, 4320, 145 all are divisible by 5 because their ones digit is 5 or 0.<\/td>\n<\/tr>\n<tr>\n<td width=\"97\">\n<p style=\"text-align: center;\">6<\/p>\n<\/td>\n<td width=\"272\">A number is divisible by 6, if the number is divisible by 2 and 3.<\/td>\n<td width=\"302\">In 5922, ones digit is 2, so it is divisible by 2. The sum of digits in 5922 is 5 + 9 + 2 + 2 = 18, which is divisible by 3. So, 5922 is divisible by 6.<\/td>\n<\/tr>\n<tr>\n<td width=\"97\">\n<p style=\"text-align: center;\">7<\/p>\n<\/td>\n<td width=\"272\">A number is divisible by 7, if the difference between twice the last digit and the number formed by other digits is either 0 or a multiple of 7.<\/td>\n<td width=\"302\">In number 2975, it is observed that the last digit in 2975 is 5. So, 297 &#8211; (2 x 5) = 287, which is a multiple of 7. Hence, 2975 is divisible by 7.<\/td>\n<\/tr>\n<tr>\n<td width=\"97\">\n<p style=\"text-align: center;\">\u00a08<\/p>\n<\/td>\n<td width=\"272\">A number is divisible by 8, if the number formed by its last three digits is divisible by 8.<\/td>\n<td width=\"302\">In 213456, the last three digits are 456 which is divisible by 8. So, the number 213456 is divisible by 8.<\/td>\n<\/tr>\n<tr>\n<td width=\"97\">\n<p style=\"text-align: center;\">9<\/p>\n<\/td>\n<td width=\"272\">A number is divisible by 9, if the sum of its digits is divisible by 9.<\/td>\n<td width=\"302\">In 538425, the sum of the digits are (5 + 3 + 8 + 4 + 2 + 5) = 27 which is divisible by 9. So, 538425 is divisible by 9.<\/td>\n<\/tr>\n<tr>\n<td width=\"97\">\n<p style=\"text-align: center;\">10<\/p>\n<\/td>\n<td width=\"272\">A number is divisible by 10, if the digit at ones place of the number is 0.<\/td>\n<td width=\"302\">The numbers 980, 63990 are all divisible by 10 because their ones digit is 0.<\/td>\n<\/tr>\n<tr>\n<td width=\"97\">\n<p style=\"text-align: center;\">11<\/p>\n<p>&nbsp;<\/td>\n<td width=\"272\">A number is divisible by 11, if the difference between the sum of digits at odd places and the sum of digits at even places is either 0 or a multiple of 11.<\/td>\n<td width=\"302\">In number 27896, the sum of the digits at odd places are (2 + 8 + 6) = 16. The sum of the digits at even places are (7 + 9 ) = 16. Their difference is 16 &#8211; 16 = 0. So, the number 27896 is divisible by 11.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><strong>Example 3:<\/strong> Test whether 72148 is divisible by 8 or not?<br \/>\n<strong>Solution:<\/strong> Here, the number formed by the last three digits is 148, which is not divisible by 8.<br \/>\nSo, 72148 is not divisible by 8.<\/p>\n<p><strong>Example 4:<\/strong> Test whether 8050314052 is divisible by 11 or not?<br \/>\n<strong>Solution:<\/strong> The sum of the digits at even places = 8 + 5 + 3 + 4 + 5 = 25<br \/>\nThe sum of digits at the odd places = 0 + 0 + 1 + 0 + 2 = 3<br \/>\nDifference = 25 &#8211; 3 = 22 22 is divisible by 11.<br \/>\nSo, the number 8050314052 is divisible by 11.<\/p>\n<p><a class=\"button-red\" title=\"Maths\" href=\"https:\/\/www.aplustopper.com\/maths\/\">Maths<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>FACTORS AND MULTIPLES Mr Sharma wanted to withdraw Rs. 1000 from his\u00a0bank account to purchase books for his children.\u00a0The cashier gave him 10 hundred-rupee notes, i.e., Rs. 10 \u00d7 100 = Rs. 1000 Mr Sharma got the required amount. But the cashier could also give the same amount in the following ways: Here, we observe [&hellip;]<\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_genesis_hide_title":false,"_genesis_hide_breadcrumbs":false,"_genesis_hide_singular_image":false,"_genesis_hide_footer_widgets":false,"_genesis_custom_body_class":"","_genesis_custom_post_class":"","_genesis_layout":"","footnotes":""},"categories":[5],"tags":[2362,31,2356,2361,55912,2357,55948,2360,2358,2359],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v17.8 (Yoast SEO v17.8) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Finding Factors And Multiples - A Plus Topper<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.aplustopper.com\/finding-factors-multiples\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Finding Factors And Multiples\" \/>\n<meta property=\"og:description\" content=\"FACTORS AND MULTIPLES Mr Sharma wanted to withdraw Rs. 1000 from his\u00a0bank account to purchase books for his children.\u00a0The cashier gave him 10 hundred-rupee notes, i.e., Rs. 10 \u00d7 100 = Rs. 1000 Mr Sharma got the required amount. 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