{"title":"Applications in Number Theory","authors":"R. Specifically, R. S T G, Sha Thb","doi":"10.1017/9781108778459.015","DOIUrl":"https://doi.org/10.1017/9781108778459.015","url":null,"abstract":"Greatest common divisor as a linear combination Theorem If a and b are positive integers and gcd(a, b) = d then there are integers s and t such that d = s×a + t×b. We illustrate first a method of finding these multipliers s and t by reversing the calculations of the Euclidean Algorithm. Later we show a direct way of finding s and t using the Extended Euclidean Algorithm. The Extended Euclidean Algorithm can be used to find the gcd of two numbers and express it as a linear combination of those numbers. It uses auxiliary numbers 1 and 0 and two starting conditions to produce an invariant expression G = S×A + T×B that yields the desired result. Example We can show that gcd(356, 252) = 4 and that 4 = (17)356 + (-24)252 In the following tableau, the first two lines express A and B as linear combinations of themselves. The calculation begins in the third line where Q n = floor (R (n–2) / R (n–1)) and A = R (-1) and B = R (0). Each of the other columns uses Q n to find the subsequent entry, and the process is repeated for each line. A 356 1 0 356 = 1HA + 0HB B 252 0 1 252 = 0HA + 1HB","PeriodicalId":385815,"journal":{"name":"Assouad Dimension and Fractal Geometry","volume":"116 1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125861257","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Two Famous Problems in Geometric Measure Theory","authors":"","doi":"10.1017/9781108778459.012","DOIUrl":"https://doi.org/10.1017/9781108778459.012","url":null,"abstract":"","PeriodicalId":385815,"journal":{"name":"Assouad Dimension and Fractal Geometry","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130138574","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Applications in Probability Theory","authors":"Prapun Suksompong","doi":"10.1017/9781108778459.016","DOIUrl":"https://doi.org/10.1017/9781108778459.016","url":null,"abstract":"","PeriodicalId":385815,"journal":{"name":"Assouad Dimension and Fractal Geometry","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116692831","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Dimensions of Measures","authors":"","doi":"10.1017/9781108778459.005","DOIUrl":"https://doi.org/10.1017/9781108778459.005","url":null,"abstract":"","PeriodicalId":385815,"journal":{"name":"Assouad Dimension and Fractal Geometry","volume":"65 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124494646","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Weak Tangents and Microsets","authors":"","doi":"10.1017/9781108778459.006","DOIUrl":"https://doi.org/10.1017/9781108778459.006","url":null,"abstract":"","PeriodicalId":385815,"journal":{"name":"Assouad Dimension and Fractal Geometry","volume":"53 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132718252","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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