What Is The Highest Power Of 2 That Divides 533! ? The Highest Power Of 3? The Highest Power Of 6<\/p>\n\n\n\n
Show That 95! Ends In 22 Zeros; With How Many Zeros Does 1001 End?<\/p>\n\n\n\n
The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n To determine the highest powers of 2, 3, and 6 that divide (533!), and the number of trailing zeros in (95!) and (1001!), we will need to apply the formula for the highest power of a prime (p) that divides (n!):<\/p>\n\n\n\n [ For powers of 2, we apply the formula for (p = 2):<\/p>\n\n\n\n [ Summing these gives the highest power of 2 in (533!).<\/p>\n\n\n\n For powers of 3, we use the same formula but with (p = 3):<\/p>\n\n\n\n [ Since (6 = 2 \\times 3), the highest power of 6 that divides (533!) is the minimum of the highest powers of 2 and 3.<\/p>\n\n\n\n The number of trailing zeros in (n!) is determined by the number of factors of 5 in (n!), because each factor of 5 pairs with a factor of 2 to form a trailing zero. The formula is:<\/p>\n\n\n\n [ For (n = 95), we compute:<\/p>\n\n\n\n [ Thus, (95!) ends in 22 zeros.<\/p>\n\n\n\n For (n = 1001), we compute the number of factors of 5 in (1001!):<\/p>\n\n\n\n [ Thus, (1001!) ends in 249 zeros.<\/p>\n\n\n\n What Is The Highest Power Of 2 That Divides 533! ? The Highest Power Of 3? The Highest Power Of 6 Show That 95! Ends In 22 Zeros; With How Many Zeros Does 1001 End? The Correct Answer and Explanation is : To determine the highest powers of 2, 3, and 6 that divide (533!), […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-180939","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/180939","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=180939"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/180939\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=180939"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=180939"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=180939"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\\text{Highest power of } p \\text{ dividing } n! = \\sum_{k=1}^{\\infty} \\left\\lfloor \\frac{n}{p^k} \\right\\rfloor
]<\/p>\n\n\n\n1. Highest Power of 2 in (533!):<\/h3>\n\n\n\n
\\left\\lfloor \\frac{533}{2} \\right\\rfloor + \\left\\lfloor \\frac{533}{4} \\right\\rfloor + \\left\\lfloor \\frac{533}{8} \\right\\rfloor + \\cdots
]<\/p>\n\n\n\n2. Highest Power of 3 in (533!):<\/h3>\n\n\n\n
\\left\\lfloor \\frac{533}{3} \\right\\rfloor + \\left\\lfloor \\frac{533}{9} \\right\\rfloor + \\left\\lfloor \\frac{533}{27} \\right\\rfloor + \\cdots
]<\/p>\n\n\n\n3. Highest Power of 6 in (533!):<\/h3>\n\n\n\n
4. Number of Zeros in (95!):<\/h3>\n\n\n\n
\\left\\lfloor \\frac{n}{5} \\right\\rfloor + \\left\\lfloor \\frac{n}{25} \\right\\rfloor + \\left\\lfloor \\frac{n}{125} \\right\\rfloor + \\cdots
]<\/p>\n\n\n\n
\\left\\lfloor \\frac{95}{5} \\right\\rfloor + \\left\\lfloor \\frac{95}{25} \\right\\rfloor + \\left\\lfloor \\frac{95}{125} \\right\\rfloor = 19 + 3 + 0 = 22
]<\/p>\n\n\n\n5. Number of Zeros in (1001!):<\/h3>\n\n\n\n
\\left\\lfloor \\frac{1001}{5} \\right\\rfloor + \\left\\lfloor \\frac{1001}{25} \\right\\rfloor + \\left\\lfloor \\frac{1001}{125} \\right\\rfloor + \\left\\lfloor \\frac{1001}{625} \\right\\rfloor = 200 + 40 + 8 + 1 = 249
]<\/p>\n\n\n\nConclusion:<\/h3>\n\n\n\n
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