{"id":23134,"date":"2022-07-24T19:57:00","date_gmt":"2022-07-24T14:27:00","guid":{"rendered":"https:\/\/e-gmat.com\/blogs\/?p=23134"},"modified":"2022-11-18T13:38:02","modified_gmt":"2022-11-18T08:08:02","slug":"properties-of-numbers-even-odd-prime-hcf-lcm","status":"publish","type":"post","link":"https:\/\/e-gmat.com\/blogs\/properties-of-numbers-even-odd-prime-hcf-lcm\/","title":{"rendered":"Properties of Numbers &#8211; Even &#038; Odd | Prime | HCF &#038; LCM"},"content":{"rendered":"<span class=\"span-reading-time rt-reading-time\" style=\"display: block;\"><span class=\"rt-label rt-prefix\">A <\/span> <span class=\"rt-time\"> 8<\/span> <span class=\"rt-label rt-postfix\">min read <\/span><\/span>\n<p class=\"wp-block-paragraph\">Hello Reader,<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">We see that you want to know the basics and properties of numbers.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><em>You have landed just at the right place.&nbsp;<\/em><\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-purpose-of-the-article\">Purpose of the article:<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">In this article,<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>You will get a deeper idea about the behavior of numbers and its operations<\/li><li>You will also get to know a few facts about the properties of the numbers.<\/li><\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">So, let\u2019s get straight into the article.<\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"576\" src=\"https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2019\/07\/Properties-of-Numbers-1024x576.jpg\" alt=\"Properties of Numbers - Even and Odd - Prime numbers - HCF and LCM\" class=\"wp-image-23139\" srcset=\"https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2019\/07\/Properties-of-Numbers-1024x576.jpg 1024w, https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2019\/07\/Properties-of-Numbers-300x169.jpg 300w, https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2019\/07\/Properties-of-Numbers-768x432.jpg 768w, https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2019\/07\/Properties-of-Numbers-1080x608.jpg 1080w, https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2019\/07\/Properties-of-Numbers.jpg 1920w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<h1 class=\"wp-block-heading\" id=\"h-properties-of-numbers-even-odd\">Properties of numbers: Even \u2013 Odd<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-even-numbers\">Even Numbers<\/h2>\n\n\n\n<ul class=\"wp-block-list\"><li>Any integer that is a multiple of 2 is an even number.<\/li><li>So, any integer which is in the form of 2k, where k is an integer, is an even number<\/li><li>So, all integers that end with 0, 2, 4, 6 or 8 are even numbers<\/li><\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><u>E\u00d7ample<\/u>: 4, 56, 98, 200 are all even numbers<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-odd-numbers\">Odd Numbers<img loading=\"lazy\" decoding=\"async\" class=\" wp-image-23137 alignright\" src=\"https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2019\/07\/Properties-of-numbers-even-and-odd-numbers-1024x768.jpg\" alt=\"Properties of numbers - even and odd numbers\" width=\"451\" height=\"338\" srcset=\"https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2019\/07\/Properties-of-numbers-even-and-odd-numbers-1024x768.jpg 1024w, https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2019\/07\/Properties-of-numbers-even-and-odd-numbers-300x225.jpg 300w, https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2019\/07\/Properties-of-numbers-even-and-odd-numbers-768x576.jpg 768w, https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2019\/07\/Properties-of-numbers-even-and-odd-numbers-510x382.jpg 510w, https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2019\/07\/Properties-of-numbers-even-and-odd-numbers-1080x810.jpg 1080w\" sizes=\"auto, (max-width: 451px) 100vw, 451px\" \/><\/h2>\n\n\n\n<ul class=\"wp-block-list\"><li>Any integer which is not an even number is an odd number<\/li><li>Or, in other words, an integer that is <em><u>not<\/u><\/em> a multiple of 2 is an odd number<\/li><li>So, any integer which is in the form of 2k \u00b1 1, where k is an integer, is an odd number<\/li><li>So, all integers that end with 1, 3, 5, 7 or 9 are odd numbers<\/li><\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><u>Example<\/u>: 7, 31, 75, 499 are all odd numbers<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Note: <\/strong>There is an odd number between every two consecutive even numbers.<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\"><p>According to the definitions given above is 0 an even or an odd number?<\/p><\/blockquote>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-properties-of-numbers-even-and-odd-number-properties\">Properties of numbers: Even and Odd number properties<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Now that we know what are even and odd numbers, let\u2019s see their properties.<\/p>\n\n\n<div class='three_fourth'>\n\t\t\t\t\t<table width=\"638\">\n<tbody>\n<tr>\n<td width=\"114\"><strong>Number 1<\/strong><\/td>\n<td width=\"41\"><strong>Operation<\/strong><\/td>\n<td width=\"100\"><strong>Number 2<\/strong><\/td>\n<td width=\"46\"><strong>\u00a0<\/strong><\/td>\n<td width=\"177\"><strong>Outcome<\/strong><\/td>\n<td width=\"160\"><strong>E<\/strong><strong>xamples<\/strong><\/td>\n<\/tr>\n<tr>\n<td width=\"114\"><span style=\"color: #008000;\"><strong>Even<\/strong><\/span><\/td>\n<td width=\"41\">\u00b1<\/td>\n<td width=\"100\"><span style=\"color: #008000;\">Even<\/span><\/td>\n<td width=\"46\">=<\/td>\n<td width=\"177\"><span style=\"color: #008000;\">Even<\/span><\/td>\n<td width=\"160\">\u00b7\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 4 + 2 = 4<\/p>\n<p>\u00b7\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 4 \u2013 2 = 2<\/td>\n<\/tr>\n<tr>\n<td width=\"114\"><span style=\"color: #008000;\"><strong>Even<\/strong><\/span><\/td>\n<td width=\"41\">\u00b1<\/td>\n<td width=\"100\"><span style=\"color: #ff9900;\">Odd<\/span><\/td>\n<td width=\"46\">=<\/td>\n<td width=\"177\"><span style=\"color: #ff9900;\">Odd<\/span><\/td>\n<td width=\"160\">\u00b7\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 4 + 1 = 5<\/p>\n<p>\u00b7\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 4 \u2013 1 = 3<\/td>\n<\/tr>\n<tr>\n<td width=\"114\"><span style=\"color: #ff9900;\"><strong>Odd<\/strong><\/span><\/td>\n<td width=\"41\">\u00b1<\/td>\n<td width=\"100\"><span style=\"color: #008000;\">Even<\/span><\/td>\n<td width=\"46\">=<\/td>\n<td width=\"177\"><span style=\"color: #ff9900;\">Odd<\/span><\/td>\n<td width=\"160\">\u00b7\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 5 + 2 = 7<\/p>\n<p>\u00b7\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 5 \u2013 2 = 3<\/td>\n<\/tr>\n<tr>\n<td width=\"114\"><span style=\"color: #ff9900;\"><strong>Odd<\/strong><\/span><\/td>\n<td width=\"41\">\u00b1<\/td>\n<td width=\"100\"><span style=\"color: #ff9900;\">Odd<\/span><\/td>\n<td width=\"46\">=<\/td>\n<td width=\"177\"><span style=\"color: #008000;\">Even<\/span><\/td>\n<td width=\"160\">\u00b7\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 5 + 3 = 8<\/p>\n<p>\u00b7\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 5 \u2013 3 = 2<\/td>\n<\/tr>\n<tr>\n<td width=\"114\"><span style=\"color: #008000;\"><strong>Even<\/strong><\/span><\/td>\n<td width=\"41\">\u00d7<\/td>\n<td width=\"100\"><span style=\"color: #008000;\">Even<\/span><\/td>\n<td width=\"46\">=<\/td>\n<td width=\"177\"><span style=\"color: #008000;\">Even<\/span><\/td>\n<td width=\"160\">\u00b7\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 2 \u00d7 4 = 8<\/td>\n<\/tr>\n<tr>\n<td width=\"114\"><span style=\"color: #008000;\"><strong>Even<\/strong><\/span><\/td>\n<td width=\"41\">\u00d7<\/td>\n<td width=\"100\"><span style=\"color: #ff9900;\">Odd<\/span><\/td>\n<td width=\"46\">=<\/td>\n<td width=\"177\"><span style=\"color: #008000;\">Even<\/span><\/td>\n<td width=\"160\">\u00b7\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 2 \u00d7 3 = 6<\/td>\n<\/tr>\n<tr>\n<td width=\"114\"><span style=\"color: #ff9900;\"><strong>Odd<\/strong><\/span><\/td>\n<td width=\"41\">\u00d7<\/td>\n<td width=\"100\"><span style=\"color: #008000;\">Even<\/span><\/td>\n<td width=\"46\">=<\/td>\n<td width=\"177\"><span style=\"color: #008000;\">Even<\/span><\/td>\n<td width=\"160\">\u00b7\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 3 \u00d7 2 = 6<\/td>\n<\/tr>\n<tr>\n<td width=\"114\"><span style=\"color: #ff9900;\"><strong>Odd<\/strong><\/span><\/td>\n<td width=\"41\">\u00d7<\/td>\n<td width=\"100\"><span style=\"color: #ff9900;\">Odd<\/span><\/td>\n<td width=\"46\">=<\/td>\n<td width=\"177\"><span style=\"color: #ff9900;\">Odd<\/span><\/td>\n<td width=\"160\">\u00b7\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 3 \u00d7 5 = 15<\/td>\n<\/tr>\n<tr>\n<td width=\"114\"><span style=\"color: #008000;\"><strong>Even<\/strong><\/span><\/td>\n<td width=\"41\">\u00f7<\/td>\n<td width=\"100\"><span style=\"color: #008000;\">Even<\/span><\/td>\n<td width=\"46\">=<\/td>\n<td width=\"177\"><span style=\"color: #ff0000;\">May or may not be an integer<\/span><\/td>\n<td width=\"160\">\u00b7\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 4 \u00f7 2 = 2<\/p>\n<p>\u00b7\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 2 \u00f7 4 = 0.5<\/td>\n<\/tr>\n<tr>\n<td width=\"114\"><span style=\"color: #008000;\"><strong>Even<\/strong><\/span><\/td>\n<td width=\"41\">\u00f7<\/td>\n<td width=\"100\"><span style=\"color: #ff9900;\">Odd<\/span><\/td>\n<td width=\"46\">=<\/td>\n<td width=\"177\"><span style=\"color: #ff0000;\">May or may not be an integer<\/span><\/td>\n<td width=\"160\">\u00b7\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 4 \u00f7 1 = 4<\/p>\n<p>\u00b7\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 4 \u00f7 3 = 1.33<\/td>\n<\/tr>\n<tr>\n<td width=\"114\"><span style=\"color: #ff9900;\"><strong>Odd<\/strong><\/span><\/td>\n<td width=\"41\">\u00f7<\/td>\n<td width=\"100\"><span style=\"color: #008000;\">Even<\/span><\/td>\n<td width=\"46\">=<\/td>\n<td width=\"177\"><span style=\"color: #ff0000;\">Not an integer<\/span><\/td>\n<td width=\"160\">\u00b7\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 3 \u00f7 2 = 1.5<\/td>\n<\/tr>\n<tr>\n<td width=\"114\"><span style=\"color: #ff9900;\"><strong>Odd<\/strong><\/span><\/td>\n<td width=\"41\">\u00f7<\/td>\n<td width=\"100\"><span style=\"color: #ff9900;\">Odd<\/span><\/td>\n<td width=\"46\">=<\/td>\n<td width=\"177\"><span style=\"color: #ff0000;\">May or may not be an integer<\/span><\/td>\n<td width=\"160\">\u00b7\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 3 \u00f7 1 = 3<\/p>\n<p>\u00b7\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 1 \u00f7 3 = 0.33<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\t\t\t\t<\/div>\n\n\n<div class='one_fourth et_column_last'>\n\t\t\t\t\t\n\t\t\t\t<\/div><div class='clear'><\/div>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-properties-of-numbers-prime-numbers\">Properties of Numbers: Prime Numbers<img loading=\"lazy\" decoding=\"async\" class=\" wp-image-23138 alignright\" src=\"https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2019\/07\/Properties-of-numbers-prime-numbers-1024x1024.jpg\" alt=\"Properties of numbers - prime numbers\" width=\"450\" height=\"450\" srcset=\"https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2019\/07\/Properties-of-numbers-prime-numbers-1024x1024.jpg 1024w, https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2019\/07\/Properties-of-numbers-prime-numbers-150x150.jpg 150w, https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2019\/07\/Properties-of-numbers-prime-numbers-300x300.jpg 300w, https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2019\/07\/Properties-of-numbers-prime-numbers-768x768.jpg 768w, https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2019\/07\/Properties-of-numbers-prime-numbers-1080x1080.jpg 1080w\" sizes=\"auto, (max-width: 450px) 100vw, 450px\" \/><\/h2>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"h-definition\">Definition<\/h3>\n\n\n\n<ul class=\"wp-block-list\"><li>A number which is divisible by only two number, 1 and the number itself, is called a prime number<\/li><\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><u>Example<\/u>: 2, 3, 5, 7, 11\u2026\u2026\u2026.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Note:<\/strong> 2 is the only prime number, which is even, because all other even numbers will be divisible by at least three numbers, 1, 2 and the number itself.<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>Every positive integer can be expressed as a product of one or more prime numbers<\/li><\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><u>Example<\/u>: 55 = 5 * 11, where 5 and 11 are two prime numbers<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\"><p>Based on the above definition, Is 1 a prime number?<\/p><\/blockquote>\n\n\n\n<p class=\"wp-block-paragraph\"><em>So, we know what is a prime number, but how do we check whether a given number is prime or not?<\/em><\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-what-is-the-process-to-check-whether-a-given-number-is-prime-or-not\">What is the process to check whether a given number is prime or not?<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">It is easy to check whether a single-digit or a two-digit number is prime or not, but what if we are given a three-digit number or more.<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li><em>Let\u2019s say the number is 123, can we quickly find whether 123 is prime or not?<\/em><\/li><\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">For this, let us learn a five-step approach to check whether a given number is a prime number or not?<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Step 1<\/strong>: <em>Find the square root of the given number<\/em><\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>The square root of 123 approximately equal to 11 as 11<sup>2<\/sup> = 121<\/li><\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Step 2<\/strong>: <em>Round it off to the closest integer<\/em><\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>The closest integer is 11.<\/li><\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Step 3<\/strong>: <em>List down all the prime numbers which are less or equal to this integer<\/em><\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>Prime numbers less than or equal to 11 are 2, 3, 5, 7, and 11.<\/li><\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Step 4<\/strong>:<em> Check whether any of these prime numbers can divide the given number or not <\/em><\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>123 is not divisible by 2.<\/li><li>But 123 is divisible by 3.<\/li><li>We need not check for other prime numbers<\/li><\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Step 5<\/strong>: <em>If yes then the given number is not a prime number, else it is a prime number<\/em><\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>123 is divisible by 3. Hence, it is not a prime number.<\/li><\/ul>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\"><p>Pop-Quiz: is 157 a prime number?<\/p><\/blockquote>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\"><p>Questions on Number Properties are very commonly asked on the GMAT. Ace GMAT Quant by <a href=\"https:\/\/resources.e-gmat.com\/sign-up-free-trial\" target=\"_blank\" rel=\"noreferrer noopener external\" data-wpel-link=\"external\">signing up for our free trial<\/a> and get access to 400+ questions. 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rel=0\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer rel=0; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"\"><\/iframe>\n\n\n<div style=\"gap: 20px\" class=\"align-button-center ub-buttons orientation-button-row ub-flex-wrap wp-block-ub-button\" id=\"ub-button-98897740-e1b7-40ca-a4dc-5c2ae6b405f9\"><div class=\"ub-button-container\">\n\t\t\t<a href=\"https:\/\/resources.e-gmat.com\/create-study-plan\" target=\"_blank\" rel=\"noopener noreferrer\" class=\"ub-button-block-main   ub-button-flex\" role=\"button\" style=\"--ub-button-background-color: #0693e3; --ub-button-color: #ffffff; --ub-button-border: none; --ub-button-hover-background-color: #cf2e2e; --ub-button-hover-color: #ffffff; --ub-button-hover-border: none; padding-top: 10px; padding-right: 10px; padding-bottom: 10px; padding-left: 10px; border-radius: 60px; \">\n\t\t\t\t<div class=\"ub-button-content-holder\" style=\"flex-direction: row\">\n\t\t\t\t\t<span class=\"ub-button-icon-holder\">\n\t\t\t\t<svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" height=\"30\" width=\"30\" viewbox=\"0 0 576 512\">\n\t\t\t\t\t<path fill=\"currentColor\" d=\"M249.6 471.5c10.8 3.8 22.4-4.1 22.4-15.5V78.6c0-4.2-1.6-8.4-5-11C247.4 52 202.4 32 144 32C93.5 32 46.3 45.3 18.1 56.1C6.8 60.5 0 71.7 0 83.8V454.1c0 11.9 12.8 20.2 24.1 16.5C55.6 460.1 105.5 448 144 448c33.9 0 79 14 105.6 23.5zm76.8 0C353 462 398.1 448 432 448c38.5 0 88.4 12.1 119.9 22.6c11.3 3.8 24.1-4.6 24.1-16.5V83.8c0-12.1-6.8-23.3-18.1-27.6C529.7 45.3 482.5 32 432 32c-58.4 0-103.4 20-123 35.6c-3.3 2.6-5 6.8-5 11V456c0 11.4 11.7 19.3 22.4 15.5z\">\n\t\t\t\t<\/path><\/svg>\n\t\t\t<\/span><span class=\"ub-button-block-btn\">Create your Personalized Study Plan<\/span>\n\t\t\t\t<\/div>\n\t\t\t<\/a>\n\t\t<\/div><\/div>\n\n\n<h2 class=\"wp-block-heading\" id=\"h-prime-factorization\">Prime factorization<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">We already know that any given positive integer can be expressed as a product of one or more prime numbers. This representation of any number is called prime factorization.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><u>For example<\/u>:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">420 = 2 \u00d7 210 = 2 \u00d7 2 \u00d7 105 = 2 \u00d7 2 \u00d7 3 \u00d7 35 = 2 \u00d7 2 \u00d7 3 \u00d7 5 \u00d7 7 = 2<sup>2<\/sup> \u00d7 3 \u00d7 5 \u00d7 7<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Now, let us learn a few more properties of numbers where we apply prime factorization.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-properties-of-numbers-least-common-multiple-lcm\">Properties of numbers: Least Common Multiple (LCM)<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Before we see what an LCM is, let us understand the meaning of a multiple.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong><u>Multiple:<\/u><\/strong> If the remainder when a number \u201cN\u201d is divided by another number \u201cn\u201d is zero, <em>then N is said to be a multiple of n<\/em>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">And, LCM is the smallest common multiple of any two or more given positive integers.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><u>For example<\/u>: LCM of 4 and 6 is 12, since 12 is the smallest number, which is a multiple of both 4 and 6<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-properties-of-numbers-highest-common-factor-or-greatest-common-divisor-hcf-gcd\">Properties of numbers: Highest Common Factor or Greatest Common Divisor (HCF\/GCD)<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Let us first learn what is a factor or divisor, and then we can learn about GCD or HCF.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong><u>Factor\/Divisor: <\/u><\/strong>If the remainder when a number \u201cN\u201d is divided by another number \u201cn\u201d is zero, then <em>n is said to be a factor or divisor of N<\/em>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">And, HCF or GCD is the largest common factor\/divisor to any two or more given positive integers.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><em><u>For example:<\/u><\/em>&nbsp;The GCD of 24 and 30 is 6, since 6 is the greatest number, which is a factor of both 24 and 30.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-method-to-find-lcm-and-gcd-of-any-two-given-positive-integers\">Method to find LCM and GCD of any two given positive integers<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Now that we know what is an LCM and GCD, let us learn a method to find the LCM and GCD of any two given positive integers.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"h-how-to-find-lcm\">How to find LCM?<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Let us consider two positive integers, 72 and 300<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Step 1<\/strong>: <em>Represent the two given numbers in their prime factorization form<\/em><\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>72 = 2<sup>3<\/sup> \u00d7 3<sup>2<\/sup><\/li><li>300 = 2<sup>2<\/sup> \u00d7 3 \u00d7 5<sup>2<\/sup><\/li><\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Step 2<\/strong>: <em>List down all the distinct prime factors from both the numbers<\/em><\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>Prime factors of 72 = 2 and 3<\/li><li>Prime factors of 300 = 2, 3 and 5<\/li><li>Thus, distinct prime factors from both combined are 2, 3 and 5<\/li><\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Step 3<\/strong>: <em>Find the power of each of these prime factors, which is the greatest between both the numbers.<\/em><\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>Power of 2 in 72 = 3, and power of 2 in 300 = 2,<\/li><li>So, the power of 2 in LCM of 72 and 300 = 3, since the power of 2 is greatest in 72<\/li><li>Similarly, we get the powers of 3 and 5 as 2 and 2 respectively<\/li><\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Step 4<\/strong>: <em>Multiply all these terms to get the LCM<\/em><\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>Therefore, the LCM of 72 and 300 = 2<sup>3<\/sup> \u00d7 3<sup>2<\/sup> \u00d7 5<sup>2<\/sup><\/li><\/ul>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"h-how-to-find-gcd-hcf\">How to find GCD\/HCF?<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Let us consider the same two positive integers, 72 and 300<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Step 1<\/strong>: <em>Represent the two given numbers in their prime factorization form<\/em><\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>72 = 2<sup>3<\/sup> \u00d7 3<sup>2<\/sup><\/li><li>300 = 2<sup>2<\/sup> \u00d7 3 \u00d7 5<sup>2<\/sup><\/li><\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Step 2<\/strong>: <em>List down all the common prime factors<\/em><\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>Prime factors of 72 = 2 and 3<\/li><li>Prime factors of 300 = 2, 3 and 5<\/li><li>Thus, common prime factors 2 and 3<\/li><\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Step 3<\/strong>: <em>Find the power of each of these prime factors, which is the smallest between both<\/em><\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>Power of 2 in 72 = 3, and power of 2 in 300 = 2,<\/li><li>So, the power of 2 in GCD of 72 and 300 = 2, since the power of 2 is the smallest in 300<\/li><li>Similarly, we get the powers of 3 and 5 as 1 and 0 respectively<\/li><\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Step 4<\/strong>: <em>Multiply all these terms to get the GCD<\/em><\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>Therefore, the GCD of 72 and 300 = 2<sup>2<\/sup> \u00d7 3 = 12<\/li><\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong><u>Note:<\/u><\/strong> <em>The product of LCM and GCD of any two given positive integers is equal to the product of the two given integers.<\/em><\/p>\n\n\n\n<ul class=\"wp-block-list\"><li><em>LCM \u00d7 HCF = Product of the integers<\/em><\/li><\/ul>\n\n\n\n<blockquote><p>What is the LCM and GCD of two prime numbers? If you liked this article here are a couple of more article related to number properties:<\/p><ul>\n<li><a href=\"https:\/\/e-gmat.com\/blogs\/gmat-quant-3-mistakes-in-even-odd-number-properties-questions\/\" data-wpel-link=\"internal\" target=\"_blank\" rel=\"noopener noreferrer\">3 Most common mistakes in GMAT even-odd questions<\/a><\/li>\n<li><a href=\"https:\/\/e-gmat.com\/blogs\/gmat-number-properties-3-mistakes-in-lcm-gcd-questions\/\" data-wpel-link=\"internal\" target=\"_blank\" rel=\"noopener noreferrer\">3 Most common mistakes in GMAT LCM\/GCD questions<\/a><\/li>\n<\/ul><\/blockquote>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-properties-of-numbers-application-quiz\"><span style=\"color: #333333; font-size: 30px;\">Properties of Numbers: Application Quiz<img loading=\"lazy\" decoding=\"async\" class=\" wp-image-23141 alignright\" src=\"https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2019\/07\/Properties-of-numbers-Number-properties-quiz-1024x731.jpg\" alt=\"Properties of numbers - Number properties quiz\" width=\"600\" height=\"428\" srcset=\"https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2019\/07\/Properties-of-numbers-Number-properties-quiz-1024x731.jpg 1024w, https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2019\/07\/Properties-of-numbers-Number-properties-quiz-300x214.jpg 300w, https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2019\/07\/Properties-of-numbers-Number-properties-quiz-768x549.jpg 768w, https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2019\/07\/Properties-of-numbers-Number-properties-quiz-400x284.jpg 400w, https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2019\/07\/Properties-of-numbers-Number-properties-quiz-1080x771.jpg 1080w\" sizes=\"auto, (max-width: 600px) 100vw, 600px\" \/><\/span><\/h2>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"h-question-1\">Question 1<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">What is the even\/odd nature of the expression, 2k<sup>2<\/sup> + 14k + 7, where k is a positive integer?<\/p>\n\n\n\n<ol class=\"wp-block-list\"><li>Even<\/li><li>Odd<\/li><li>Cannot be determined<\/li><\/ol>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"h-solution\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Given<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>An e\u00d7pression, 2k<sup>2<\/sup> + 14k + 7, where k is a positive integer<\/li><\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>To find<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>The even\/odd nature of the given e\u00d7pression<\/li><\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Approach and Working out<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>We can write, 2k<sup>2<\/sup> + 14k + 7 = 2k<sup>2<\/sup> + 14k + 6 + 1<\/li><\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Now, if we see, in the given e\u00d7pression<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>The term 2k<sup>2<\/sup> is always even, for any value of k<\/li><li>Since, even * any integer = even<\/li><li>The term 14k is always even, for any value of k<\/li><li>Since, even * any integer = even<\/li><li>The term 6 is also even, and 1 is odd<\/li><\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">So, 2k<sup>2<\/sup> + 14k + 7 = 2k<sup>2<\/sup> + 14k + 6 + 1 = even + even + even + odd = odd<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore, the given expression is always odd, for any value of k<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Hence, the correct answer is <strong>Option B<\/strong>.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-properties-of-numbers-quiz-question-2\">Properties of Numbers Quiz: Question 2<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">The LCM of two positive integers, p, and q, is 42 and HCF is 1. If p is an even prime number, then what is the value of q?<\/p>\n\n\n\n<ol class=\"wp-block-list\"><li>6<\/li><li>7<\/li><li>14<\/li><li>21<\/li><li>42<\/li><\/ol>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"h-solution-1\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Given<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>Two positive integers, p, and q<\/li><li>p is an even prime number<\/li><li>LCM pf p and q = 42<\/li><li>HCF of p and q = 1<\/li><\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>To find<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>The value of q<\/li><\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Approach and Working out<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The value of p = 2, since 2 is the only even prime number<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">And, we know that,<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>LCM \u00d7 HCF = the product of the two integers<\/li><li>42 \u00d7 1 = 2 \u00d7 q<\/li><\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore, the value of q = 42\/2 = 21.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Hence, the correct answer is <strong>Option D<\/strong>.<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\"><p><strong>If you are planning to take the GMAT, we can help you with a personalized study plan and give you access to quality online content to prepare.<\/strong>&nbsp;<strong>Write to us at&nbsp;<a href=\"mailto:acethegmat@e-gmat.com\" target=\"_blank\" rel=\"noreferrer noopener\">acethegmat@e-gmat.com<\/a>. 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You have landed just at the right place.&nbsp; Purpose of the article: In this article, You will get a deeper idea about the behavior of numbers and its operations You will also get to know a few facts about the [&hellip;]<\/p>\n","protected":false},"author":102412,"featured_media":23139,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_et_pb_use_builder":"","_et_pb_old_content":"","_et_gb_content_width":"","footnotes":""},"categories":[94,60,61],"tags":[],"class_list":["post-23134","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-gmat-prep","category-gmat-quant","category-number-properties","et-has-post-format-content","et_post_format-et-post-format-standard"],"featured_image_src":"https:\/\/e-gmat.com\/blogs\/wp-content\/uploads\/2019\/07\/Properties-of-Numbers.jpg","author_info":{"display_name":"Asif M","author_link":"https:\/\/e-gmat.com\/blogs\/author\/asife-gmat-com\/"},"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.2 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Properties of Numbers - Even &amp; 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