Three students need to produce a prime factorization of 48. Donna states that the first factors in the tree should be 6 and 8. Larry states that the first factors in the tree should be 4 and 12. Trish states that the initial factors of 48 do not affect the prime factorization. Explain why Trish is correct. <\/p>\n\n\n\n
The correct answer and explanation is:<\/strong><\/mark><\/p>\n\n\n\n Correct Answer:<\/strong> Prime factorization is the process of breaking down a number into a product of prime numbers only<\/strong>. Prime numbers are numbers greater than 1 that have no other factors except 1 and themselves (e.g., 2, 3, 5, 7, 11, etc.).<\/p>\n\n\n\n Let\u2019s consider the number 48<\/strong>.<\/p>\n\n\n\n Different people can start a factor tree in different ways:<\/p>\n\n\n\n Both Donna and Larry ended up with the same prime factors<\/strong>, even though they began with different pairs of factors.<\/p>\n\n\n\n Trish is absolutely correct<\/strong> because the order or method<\/strong> used to factor a number does not<\/strong> change the result of its prime factorization<\/strong>. No matter where you start, as long as you correctly break down the number into prime numbers<\/strong>, the final product will always be the same.<\/p>\n\n\n\n This is due to the Fundamental Theorem of Arithmetic<\/strong>, which states that every integer greater than 1 can be written as a unique product of prime numbers<\/strong>, regardless of the order in which they are factored.<\/p>\n\n\n\n So, while different factor trees may look different, they all lead to the same prime factorization<\/strong>: \u2705 Conclusion:<\/strong> Trish is right\u2014the starting point doesn\u2019t matter; the prime factorization will always be the same.<\/p>\n","protected":false},"excerpt":{"rendered":" Three students need to produce a prime factorization of 48. Donna states that the first factors in the tree should be 6 and 8. Larry states that the first factors in the tree should be 4 and 12. Trish states that the initial factors of 48 do not affect the prime factorization. Explain why Trish […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-27219","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/27219","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=27219"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/27219\/revisions"}],"predecessor-version":[{"id":27220,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/27219\/revisions\/27220"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=27219"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=27219"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=27219"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
Trish is correct. The initial factors chosen in a factor tree do not<\/strong> affect the final prime factorization<\/strong> of a number.<\/p>\n\n\n\n
\n\n\n\nExplanation (Approximately 300 Words):<\/strong><\/h3>\n\n\n\n
\n
48 = 6 \u00d7 8
Then:
6 = 2 \u00d7 3
8 = 2 \u00d7 2 \u00d7 2
So, 48 = 2 \u00d7 3 \u00d7 2 \u00d7 2 \u00d7 2
Rearranged: 48 = 2\u2074 \u00d7 3<\/strong><\/li>\n\n\n\n
48 = 4 \u00d7 12
Then:
4 = 2 \u00d7 2
12 = 3 \u00d7 4 = 3 \u00d7 2 \u00d7 2
So, 48 = 2 \u00d7 2 \u00d7 3 \u00d7 2 \u00d7 2
Rearranged: 48 = 2\u2074 \u00d7 3<\/strong><\/li>\n<\/ul>\n\n\n\n
48 = 2 \u00d7 2 \u00d7 2 \u00d7 2 \u00d7 3 = 2\u2074 \u00d7 3<\/strong><\/p>\n\n\n\n