{"id":1684,"date":"2012-06-05T16:45:33","date_gmt":"2012-06-05T23:45:33","guid":{"rendered":"https:\/\/magoosh.com\/gmat\/?p=1684"},"modified":"2020-01-15T10:50:41","modified_gmt":"2020-01-15T18:50:41","slug":"advanced-non-calculator-factoring-on-the-gmat","status":"publish","type":"post","link":"https:\/\/magoosh.com\/gmat\/advanced-non-calculator-factoring-on-the-gmat\/","title":{"rendered":"Advanced (Non-Calculator!) Factoring on the GMAT"},"content":{"rendered":"<p><strong>Fact:<\/strong> On the GMAT Quantitative Section, <a title=\"Can I Use a Calculator on the GMAT?\" href=\"\/\/magoosh.com\/gmat\/can-you-use-a-calculator-on-the-gmat\/\" target=\"_blank\" rel=\"noopener noreferrer\">you cannot use a calculator<\/a>.<\/p>\n<p>Therefore, if you are aiming to tackle even the 700-800 questions on the Q section, you need to have some crafty mental math tricks up your sleeve. One of the most powerful involves the clever use of a famous algebra formula: the difference of two squares formula. See <a href=\"https:\/\/magoosh.com\/gmat\/gmat-quant-difference-of-two-squares\/\" target=\"_blank\" rel=\"noopener noreferrer\">this post<\/a>\u00a0for uses of that formula in general problem solving. Here we will focus on factoring.<\/p>\n<p>&nbsp;<\/p>\n<h2>Factoring Big Numbers<\/h2>\n<p>In general, factoring a big number can be time-consuming without a calculator. The GMAT might ask a question in which you need to know the factors, or the prime factorization, of a large number. See <a href=\"https:\/\/magoosh.com\/gmat\/gmat-math-factors\/\" target=\"_blank\" rel=\"noopener noreferrer\">this post<\/a>\u00a0for more on prime factorizations.<br \/>\nFirst of all, notice how easy it is to square the multiples of 10. If you can square the numbers from 1 to 10, you can square the multiples of 10 from 10 to 100.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_993_4767c2a46b6df4197d8dfc15fb8a7cef.png\" style=\"vertical-align:-7px; display: inline-block ;\" alt=\"10^2 = 100\" title=\"10^2 = 100\"\/><br \/>\n<img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_993_ab76231de79dd9b93e414c5ccbf2c5e4.png\" style=\"vertical-align:-7px; display: inline-block ;\" alt=\"20^2 = 400\" title=\"20^2 = 400\"\/><br \/>\n<img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_993_136f0122dc4be76c0d1504fe7095dec4.png\" style=\"vertical-align:-7px; display: inline-block ;\" alt=\"30^2 = 900\" title=\"30^2 = 900\"\/><br \/>\n<img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_993_bc95dc8d63e37ab08c7835ab6442b4bf.png\" style=\"vertical-align:-7px; display: inline-block ;\" alt=\"40^2 = 1600\" title=\"40^2 = 1600\"\/><br \/>\n<img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_993_e46c54e138e9c8811011c2e5b14fd596.png\" style=\"vertical-align:-7px; display: inline-block ;\" alt=\"50^2 = 2500\" title=\"50^2 = 2500\"\/><br \/>\n<img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_993_3bbe06163a5afc8fe82833d74e0f669d.png\" style=\"vertical-align:-7px; display: inline-block ;\" alt=\"60^2 = 3600\" title=\"60^2 = 3600\"\/><br \/>\n<img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_993_9e4d119c22e3e78243ec333b32cf3eed.png\" style=\"vertical-align:-7px; display: inline-block ;\" alt=\"70^2 = 4900\" title=\"70^2 = 4900\"\/><br \/>\n<img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_993_852a9958218bed244ee95b7009e43670.png\" style=\"vertical-align:-7px; display: inline-block ;\" alt=\"80^2 = 6400\" title=\"80^2 = 6400\"\/><br \/>\n<img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_993_8e3ac0694fa752d154f26c4577c35420.png\" style=\"vertical-align:-7px; display: inline-block ;\" alt=\"90^2 = 8100\" title=\"90^2 = 8100\"\/><br \/>\n<img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_993_c400760f71a3322205c806a2504dd2ac.png\" style=\"vertical-align:-7px; display: inline-block ;\" alt=\"100^2 = 10000\" title=\"100^2 = 10000\"\/><\/p>\n<p>Those should all be recognizable as nice round perfect squares.<br \/>\nNow, suppose you are in a situation in which you have to factor, say, 1591, or 3551, or 8099. Notice, all of those are a perfect square less than one of these multiples of ten squared.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_990.5_966189e13e02f694bcded97e74cf8721.png\" style=\"vertical-align:-9.5px; display: inline-block ;\" alt=\"1591 = 1600 - 9 = (40)^2 - (3)^2 = (40 + 3)(40 - 3) = 43*37\" title=\"1591 = 1600 - 9 = (40)^2 - (3)^2 = (40 + 3)(40 - 3) = 43*37\"\/><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_990.5_e4c5f6749e2fc6414642bf03e8b38ae9.png\" style=\"vertical-align:-9.5px; display: inline-block ;\" alt=\"3551 = 3600 - 49 = (60)^2 - (7)^2 = (60 + 7)(60 - 7) = 67*53\" title=\"3551 = 3600 - 49 = (60)^2 - (7)^2 = (60 + 7)(60 - 7) = 67*53\"\/><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_990.5_45e3c10b8a883640f0fa47b1d8a98e6b.png\" style=\"vertical-align:-9.5px; display: inline-block ;\" alt=\"8099 = 8100 - 1 = (90)^2 - (1)^2 = (90 + 1)(90 - 1) = 91*89 = 7*13*89\" title=\"8099 = 8100 - 1 = (90)^2 - (1)^2 = (90 + 1)(90 - 1) = 91*89 = 7*13*89\"\/><\/p>\n<p>In general, the GMAT is not going to put you in a situation in which you have to find the prime factorization of a general four digit number. If this situation does arise, you can bet there&#8217;s an enormously simplifying trick available, and factoring via the difference of two squares is an awfully likely candidate for that trick.<\/p>\n<p>&nbsp;<\/p>\n<h2>Factoring Decimals<\/h2>\n<p>Just as the difference of two square can simplify factoring big numbers, it can also simplify factoring decimals. To demonstrate this, I am going to show the solution to a flamboyantly recondite question from the OG13e, viz., PS #199 on p. 180.<\/p>\n<p>199) <img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_984_b3f76c6a4b12e38f251f8b8c61786936.png\" style=\"vertical-align:-16px; display: inline-block ;\" alt=\"0.99999999\/1.0001 - 0.99999991\/1.0003=\" title=\"0.99999999\/1.0001 - 0.99999991\/1.0003=\"\/><br \/>\n(A) <img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_993_b3915c78b2597644b3009e0856cae815.png\" style=\"vertical-align:-7px; display: inline-block ;\" alt=\"10^-8\" title=\"10^-8\"\/><br \/>\n(B) <img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_983.5_61344adfdeffec61f1917fbfd6e4f419.png\" style=\"vertical-align:-16.5px; display: inline-block ;\" alt=\"3(10^-8)\" title=\"3(10^-8)\"\/><br \/>\n(C) <img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_983.5_2dca4570c38e6ed92e683f18c18334c0.png\" style=\"vertical-align:-16.5px; display: inline-block ;\" alt=\"3(10^-4)\" title=\"3(10^-4)\"\/><br \/>\n(D)\u00a0<img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_983.5_27476f449353e695986285217f168bb1.png\" style=\"vertical-align:-16.5px; display: inline-block ;\" alt=\"2(10^-4)\" title=\"2(10^-4)\"\/><br \/>\n(E) <img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_993_dd49cefed2b9c97ed3f94b9f6dddb7f7.png\" style=\"vertical-align:-7px; display: inline-block ;\" alt=\"10^-4\" title=\"10^-4\"\/><\/p>\n<p>Notice that <img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_983.5_a791a2d6d8bb27d946303a7b75bcce27.png\" style=\"vertical-align:-16.5px; display: inline-block ;\" alt=\"1.0001 = 1 + (10^-4)\" title=\"1.0001 = 1 + (10^-4)\"\/>, and <img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_983.5_5f0728425cc154bbbe8723b3d3e48909.png\" style=\"vertical-align:-16.5px; display: inline-block ;\" alt=\"1.0003 = 1 + 3(10^-4)\" title=\"1.0003 = 1 + 3(10^-4)\"\/>.<\/p>\n<p>Now, notice that both numerators simplify via difference of two squares formula.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_981_a88ea87ba9c246bae602842ebf25c324.png\" style=\"vertical-align:-19px; display: inline-block ;\" alt=\"0.99999999 = 1-10^(-8) = (1)^2-(10^(-4) )^2 =(1+10^(-4) )(1-10^(-4) )\" title=\"0.99999999 = 1-10^(-8) = (1)^2-(10^(-4) )^2 =(1+10^(-4) )(1-10^(-4) )\"\/><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_978.5_cee686f7892f0bd0d3f2d42f51fd709f.png\" style=\"vertical-align:-21.5px; display: inline-block ;\" alt=\"0.99999991 = 1-9(10^(-8) ) = (1)^2-(3(10^(-4) ) )^2 = (1+3(10^(-4) ) )(1-3(10^(-4) ) )\" title=\"0.99999991 = 1-9(10^(-8) ) = (1)^2-(3(10^(-4) ) )^2 = (1+3(10^(-4) ) )(1-3(10^(-4) ) )\"\/><\/p>\n<p>Thus, the two fractions become<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_952_5fb2102aed5640d993b64890cef798e4.png\" style=\"vertical-align:-48px; display: inline-block ;\" alt=\"0.99999999\/1.0001 = (1+10^(-4) )(1-10^(-4) )\/((1+10^(-4) ) ) = 1 - 10^-4\" title=\"0.99999999\/1.0001 = (1+10^(-4) )(1-10^(-4) )\/((1+10^(-4) ) ) = 1 - 10^-4\"\/><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_947_34722e0535855ef0da3ba9975b71b775.png\" style=\"vertical-align:-53px; display: inline-block ;\" alt=\"0.99999991\/1.0003 =(1+3(10^(-4) ) )(1-3(10^(-4) ) )\/((1+3(10^(-4) ) ) ) = 1 - 3(10^-4)\" title=\"0.99999991\/1.0003 =(1+3(10^(-4) ) )(1-3(10^(-4) ) )\/((1+3(10^(-4) ) ) ) = 1 - 3(10^-4)\"\/><\/p>\n<p>The difference:<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_981_885beef8a22e2e0d5516cf2b4ca1bb8e.png\" style=\"vertical-align:-19px; display: inline-block ;\" alt=\"(1 - 10^-4)- (1 - 3(10^-4)) = 3(10^-4)-10^-4 = 2(10^-4)\" title=\"(1 - 10^-4)- (1 - 3(10^-4)) = 3(10^-4)-10^-4 = 2(10^-4)\"\/><\/p>\n<p>Answer = D<\/p>\n<p>&nbsp;<\/p>\n<p>Here&#8217;s a related practice question.<\/p>\n<p><a href=\"http:\/\/gmat.magoosh.com\/questions\/129\" target=\"_blank\" rel=\"noopener noreferrer\">http:\/\/gmat.magoosh.com\/questions\/129<\/a><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Fact: On the GMAT Quantitative Section, you cannot use a calculator. Therefore, if you are aiming to tackle even the 700-800 questions on the Q section, you need to have some crafty mental math tricks up your sleeve. One of the most powerful involves the clever use of a famous algebra formula: the difference of [&hellip;]<\/p>\n","protected":false},"author":26,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[150],"tags":[],"ppma_author":[13209],"class_list":["post-1684","post","type-post","status-publish","format-standard","hentry","category-basics"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v21.7 (Yoast SEO v21.7) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>GMAT: Factoring without a Calculator | Magoosh Study Resources<\/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:\/\/magoosh.com\/gmat\/advanced-non-calculator-factoring-on-the-gmat\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Advanced (Non-Calculator!) 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He was also featured as \"member of the month\" for over two years at <a href=\"https:\/\/gmatclub.com\/blog\/2012\/09\/mike-mcgarrys-gmat-experience\/\" rel=\"noopener noreferrer\">GMAT Club<\/a>. Mike holds an A.B. in Physics (graduating <em>magna cum laude<\/em>) and an M.T.S. in Religions of the World, both from Harvard. Beyond standardized testing, Mike has over 20 years of both private and public high school teaching experience specializing in math and physics. In his free time, Mike likes smashing foosballs into orbit, and despite having no obvious cranial deficiency, he insists on rooting for the NY Mets. 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