ASAL SAYILAR
\n Asal say\u0131lar, 1 ve kendisinden ba\u015fka pozitif tam b\u00f6leni olmayan 1′ den b\u00fcy\u00fck tamsay\u0131lard\u0131r. En k\u00fc\u00e7\u00fck asal say\u0131, 2′ dir. 2 asal say\u0131s\u0131 d\u0131\u015f\u0131nda \u00e7ift asal say\u0131 yoktur. Yani, 2 say\u0131s\u0131 d\u0131\u015f\u0131ndaki t\u00fcm asal say\u0131lar tek say\u0131d\u0131r. Asal say\u0131lar k\u00fcmesi,
\n { 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, … }
\n dir.
\n Fermat Teoremi’ ne g\u00f6re, n asal say\u0131 olmak \u00fczere, 2n – 1 \u015feklinde yaz\u0131labilen say\u0131lar asal say\u0131d\u0131r. \u00d6rne\u011fin,
\n 22 – 1, 23 – 1, 25 – 1, 27 – 1, 211 – 1, …
\n say\u0131lar\u0131, asal say\u0131d\u0131r.
\n Aralar\u0131nda asal say\u0131lar:
\n 1′ den ba\u015fka pozitif ortak b\u00f6leni olmayan say\u0131lara, aralar\u0131nda asal say\u0131lar ad\u0131 verilir. Birden fazla say\u0131n\u0131n aralar\u0131nda asal olmas\u0131 i\u00e7in, bu say\u0131lar\u0131n asal say\u0131 olmas\u0131 gerekmez. Asal say\u0131lar, kesinlikle aralar\u0131nda asal say\u0131lard\u0131r. Bununla birlikte, 10 ve 81 say\u0131s\u0131 birer asal say\u0131 olmamas\u0131na ra\u011fmen, aralar\u0131nda asal say\u0131lard\u0131r. Di\u011fer taraftan, 10 ile 8 say\u0131s\u0131 birer asal say\u0131 olmamas\u0131na ra\u011fmen, 2 ortak b\u00f6lenleri oldu\u011fu i\u00e7in, aralar\u0131nda asal say\u0131lar de\u011fildir. Bir say\u0131 aralar\u0131nda asal iki say\u0131ya b\u00f6l\u00fcnebiliyorsa, bu iki say\u0131n\u0131n \u00e7arp\u0131m\u0131na da b\u00f6l\u00fcn\u00fcr.
\n \u00d6rne\u011fin,\u2022 2, 9
\n \u2022 10, 81
\n \u2022 5, 29
\n \u2022 3, 8
\n \u2022 2, 10, 35
\n say\u0131 gruplar\u0131, ortak tam b\u00f6lenleri olmad\u0131\u011f\u0131 i\u00e7in aralar\u0131nda asal say\u0131lard\u0131r.
\n Asal olmayan say\u0131lara da bile\u015fik say\u0131 ad\u0131 verilir. Dolay\u0131s\u0131yla, bile\u015fik say\u0131lar\u0131n 1 ve kendisinden ba\u015fka b\u00f6lenleri vard\u0131r. \u00d6rne\u011fin, 10 say\u0131s\u0131 bir bile\u015fik say\u0131d\u0131r. \u00c7\u00fcnk\u00fc, 10 say\u0131s\u0131n\u0131n 1 ve kendisinden ba\u015fka, 2 ile 5 b\u00f6leni vard\u0131r. Buradan, asal olmayan 10 say\u0131s\u0131, birer asal say\u0131 olan 2 say\u0131s\u0131 ile 5 say\u0131s\u0131n\u0131n \u00e7arp\u0131m\u0131 olarak yaz\u0131labilir. 2 ile 5 say\u0131s\u0131na, 10 say\u0131s\u0131n\u0131n asal \u00e7arpan\u0131 veya b\u00f6leni denir. Yani, bile\u015fik bir say\u0131, asal say\u0131lar\u0131n \u00e7arp\u0131m\u0131 \u015feklinde yaz\u0131labilir.
\n \u00d6rnek 1:
\n A\u015fa\u011f\u0131daki say\u0131 gruplar\u0131ndan hangisi aralar\u0131nda asald\u0131r?
\n a) 4, 20 b) 6, 21 c) 27, 36, 39 d) 8, 24, 36 e) 3, 5, 25
\n \u00c7\u00f6z\u00fcm:
\n a) 4 ile 20′ nin ortak b\u00f6leni vard\u0131r ve bu da 2 ile 4′ t\u00fcr.
\n b) 6 ile 21′ in ortak b\u00f6leni vard\u0131r ve bu da 3′ t\u00fcr.
\n c) 27, 36 ve 39′ un ortak b\u00f6leni vard\u0131r ve ortak b\u00f6len 3′ t\u00fcr.
\n d) 8, 24 ve 36′ n\u0131n ortak b\u00f6leni vard\u0131r ve ortak b\u00f6len 2 ve 4′ t\u00fcr.
\n e) 3, 5 ve 25′ in ortak b\u00f6leni yoktur. \u00c7\u00fcnk\u00fc, bu \u00fc\u00e7 say\u0131y\u0131 birden b\u00f6len 1′ den ba\u015fka say\u0131 yoktur. Dolay\u0131s\u0131yla, bu say\u0131lar aralar\u0131nda asald\u0131r.
\n \u00d6rnek 2:
\n 2m + 3 ile 7n – 5 say\u0131lar\u0131 aralar\u0131nda asal oldu\u011funa g\u00f6re,<\/p>\n
ise, m ve n ka\u00e7t\u0131r?<\/p>\n
\u00c7\u00f6z\u00fcm:
\n 2m + 3 ile 7n – 5 aralar\u0131nda asal olduklar\u0131na g\u00f6re,
\n 2m + 3 = 5
\n 2m = 5 – 3
\n 2m = 2
\n m = 1<\/p>\n
7n – 5 = 9
\n 7n = 9 + 5
\n 7n = 14
\n n = 2
\n bulunur.
\n \u00d6rnek 3:
\n a, b ve c birbirinden farkl\u0131 rakamlar olmak \u00fczere, ab ile bc iki basamakl\u0131 aralar\u0131nda asal say\u0131lard\u0131r. Buna g\u00f6re, ab + bc toplam\u0131n\u0131n en k\u00fc\u00e7\u00fck de\u011feri ka\u00e7t\u0131r?
\n \u00c7\u00f6z\u00fcm:
\n Toplam\u0131n en k\u00fc\u00e7\u00fck olmas\u0131 i\u00e7in, say\u0131lar\u0131 en k\u00fc\u00e7\u00fck almal\u0131y\u0131z. Buna g\u00f6re, ab = 21 olurken. bc = 13 olmal\u0131d\u0131r. Dolay\u0131s\u0131yla,
\n ab + bc = 21 + 13 = 34
\n olur.
\n \u00d6rnek 4:
\n 2x + y ile 4 x + y say\u0131lar\u0131 aralar\u0131nda asal oldu\u011funa g\u00f6re,<\/p>\n
ise, 3x + 2y toplam\u0131 ka\u00e7t\u0131r?
\n \u00c7\u00f6z\u00fcm:
\n 2x + y ile 4x + y say\u0131lar\u0131 aralar\u0131nda asal oldu\u011funa g\u00f6re, her ikisinin de ortak b\u00f6leni olmamas\u0131 gerekti\u011finden, e\u015fitli\u011fin sa\u011f taraf\u0131 ortak b\u00f6lenden ar\u0131nd\u0131r\u0131lmal\u0131d\u0131r. Dolay\u0131s\u0131yla,<\/p>\n
olur ve buradan,
\n 2x + y = 7 … (1)
\n 4x + y = 9 … (2)
\n yaz\u0131l\u0131r. Bu denklemleri ortak olarak \u00e7\u00f6zelim. Bunun i\u00e7in, (1) nolu denklemi – 1 ile \u00e7arpal\u0131m ve (1) nolu denklemle (2) nolu denklemi taraf tarafa toplayal\u0131m.
\n – 1 \/ 2x + y = 7
\n 4x + y = 9
\n – 2x – y = – 7
\n 4x + y = 9
\n Son iki denklemin toplam\u0131
\n 2x = 2
\n x = 1
\n bulunur ve x = 1 de\u011ferini (1) nolu denklemde yerine koyal\u0131m
\n 2.1 + y = 7
\n y = 7 – 2
\n y = 5
\n bulunur. Buradan
\n 3x + 2y = 3.1 + 2.5 = 3 +10 = 13
\n olur.
\n SAYILARIN ASAL \u00c7ARPANLARINA AYRILMASI
\n Her bile\u015fik say\u0131, asal say\u0131lar\u0131n veya asal say\u0131lar\u0131n kuvvetlerinin \u00e7arp\u0131m\u0131 \u015feklinde yaz\u0131labilir. Bu i\u015flemi yapmak i\u00e7in, ilgili say\u0131n\u0131n s\u0131ras\u0131yla en k\u00fc\u00e7\u00fck asal say\u0131dan ba\u015flanarak b\u00f6l\u00fcnebilmesi ara\u015ft\u0131r\u0131l\u0131r.
\n \u00d6rnek 1:
\n 124 say\u0131s\u0131n\u0131 asal \u00e7arpanlar\u0131na ay\u0131ral\u0131m.
\n \u00c7\u00f6z\u00fcm:<\/p>\n
120 = 23 . 31. 51
\n \u00d6rnek 2:
\n 500 say\u0131s\u0131n\u0131 asal \u00e7arpanlar\u0131na ay\u0131ral\u0131m.
\n \u00c7\u00f6z\u00fcm:<\/p>\n
500 = 22 . 53
\n B\u0130R SAYMA SAYISININ TAMSAYI B\u00d6LENLER\u0130<\/p>\n
Bir sayma say\u0131s\u0131n\u0131n pozitif tamsay\u0131 b\u00f6lenlerinin say\u0131s\u0131:
\n Herhangi bir A sayma say\u0131s\u0131n\u0131n asal \u00e7arpanlar\u0131 a, b ve c olmak \u00fczere,
\n A = am . bn . cp
\n \u015feklinde asal \u00e7arpanlar\u0131na ayr\u0131lm\u0131\u015f ise, A sayma say\u0131s\u0131n\u0131n pozitif tamsay\u0131 b\u00f6lenlerinin say\u0131s\u0131,
\n ( m + 1 ) . ( n + 1 ) . ( p + 1 )
\n dir. Bu say\u0131ya, 1 ile say\u0131n\u0131n kendisi dahil edilmi\u015ftir.
\n Bir sayma say\u0131s\u0131n\u0131n t\u00fcm tamsay\u0131 b\u00f6lenlerinin say\u0131s\u0131:
\n Herhangi bir A sayma say\u0131s\u0131n\u0131n asal \u00e7arpanlar\u0131 a, b ve c olmak \u00fczere,
\n A = am . bn . cp
\n \u015feklinde asal \u00e7arpanlar\u0131na ayr\u0131lm\u0131\u015f ise, A sayma say\u0131s\u0131n\u0131n t\u00fcm tamsay\u0131 b\u00f6lenlerinin say\u0131s\u0131,
\n 2 . ( m + 1 ) . ( n + 1 ) . ( p + 1 )
\n dir. Yani, A sayma say\u0131s\u0131n\u0131n t\u00fcm tamsay\u0131 b\u00f6lenlerinin say\u0131s\u0131, pozitif b\u00f6lenlerinin say\u0131s\u0131n\u0131n 2 kat\u0131d\u0131r. Bu say\u0131ya, 1 ile say\u0131n\u0131n kendisi dahil edilmi\u015ftir.
\n Bir sayma say\u0131s\u0131n\u0131n pozitif tamsay\u0131 b\u00f6lenlerinin toplam\u0131:
\n Herhangi bir A sayma say\u0131s\u0131n\u0131n asal \u00e7arpanlar\u0131 a, b ve c olmak \u00fczere,
\n A = am . bn . cp
\n \u015feklinde asal \u00e7arpanlar\u0131na ayr\u0131lm\u0131\u015f ise, A sayma say\u0131s\u0131n\u0131n pozitif tamsay\u0131 b\u00f6lenlerinin toplam\u0131,<\/p>\n
dir. Bu toplama, 1 ile say\u0131n\u0131n kendisi dahil edilmi\u015ftir. Bir sayma say\u0131s\u0131n\u0131n t\u00fcm tamsay\u0131 b\u00f6lenlerinin toplam\u0131 ise, s\u0131f\u0131rd\u0131r.
\n Bir sayma say\u0131s\u0131n\u0131n pozitif tamsay\u0131 b\u00f6lenlerinin \u00e7arp\u0131m\u0131:
\n Herhangi bir A sayma say\u0131s\u0131n\u0131n asal \u00e7arpanlar\u0131 a, b ve c olmak \u00fczere,
\n A = am . bn . cp
\n \u015feklinde asal \u00e7arpanlar\u0131na ayr\u0131lm\u0131\u015f ise, A sayma say\u0131s\u0131n\u0131n pozitif tamsay\u0131 b\u00f6lenlerinin \u00e7arp\u0131m\u0131,<\/p>\n
dir. \u00dcss\u00fcn, A n\u0131n pozitif tamsay\u0131 b\u00f6lenlerinin say\u0131s\u0131n\u0131n yar\u0131s\u0131 oldu\u011funa dikkat ediniz.
\n \u00d6rnek 1:
\n 120 say\u0131s\u0131n\u0131n
\n a) Ka\u00e7 tane pozitif b\u00f6leni vard\u0131r?
\n b) Ka\u00e7 tane tamsay\u0131 b\u00f6leni vard\u0131r?
\n c) Pozitif b\u00f6lenlerinin toplam\u0131 ka\u00e7t\u0131r?
\n d) Pozitif b\u00f6lenlerinin \u00e7arp\u0131m\u0131 ka\u00e7t\u0131r?
\n \u00c7\u00f6z\u00fcm:
\n a) 120 say\u0131s\u0131n\u0131n asal \u00e7arpanlar\u0131na ayr\u0131lm\u0131\u015f \u015fekli
\n 120 = 23 . 31. 51
\n oldu\u011fundan, pozitif b\u00f6lenlerinin say\u0131s\u0131
\n ( 3 + 1) . ( 1 + 1 ) . ( 1 + 1 ) = 4 . 2 . 2 = 16
\n d\u0131r.
\n b) 120 say\u0131s\u0131n\u0131n t\u00fcm b\u00f6lenlerinin say\u0131s\u0131, pozitif b\u00f6lenlerinin say\u0131s\u0131n\u0131n 2 kat\u0131 oldu\u011funa g\u00f6re,
\n 2 . 16 = 32
\n dir.
\n c) 120 say\u0131s\u0131n\u0131n pozitif b\u00f6lenlerinin toplam\u0131<\/p>\n
dir.
\n d) 120 say\u0131s\u0131n\u0131n pozitif b\u00f6lenlerinin \u00e7arp\u0131m\u0131<\/p>\n
dir.
\n \u00d6rnek 2:
\n 500 . 5y say\u0131s\u0131n\u0131n asal olmayan 40 tane tamsay\u0131 b\u00f6leni varsa, y ka\u00e7t\u0131r?
\n \u00c7\u00f6z\u00fcm:
\n 500 . 5y = 22 . 53 . 5y
\n = 22 . 53 + y
\n 2 tane asal b\u00f6leni oldu\u011fundan, t\u00fcm b\u00f6lenlerinin say\u0131s\u0131,
\n 40 + 2 = 42
\n dir. Buradan, pozitif b\u00f6lenlerinin say\u0131s\u0131, t\u00fcm b\u00f6lenlerinin say\u0131s\u0131n\u0131n yar\u0131s\u0131 oldu\u011fundan,
\n 21 = ( 2 + 1 ) . ( 3 + x + 1 )
\n 21 = 3 . ( 4 + x )
\n 21 = 12 + 3x
\n 3x = 21 – 12
\n 3x = 9
\n x = 3
\n olur.<\/p>\n","protected":false},"excerpt":{"rendered":"
ASAL SAYILAR Asal say\u0131lar, 1 ve kendisinden ba\u015fka pozitif tam b\u00f6leni olmayan 1′ den b\u00fcy\u00fck tamsay\u0131lard\u0131r. En k\u00fc\u00e7\u00fck asal say\u0131, 2′ dir. 2 asal say\u0131s\u0131 d\u0131\u015f\u0131nda \u00e7ift asal say\u0131 yoktur. Yani, 2 say\u0131s\u0131 d\u0131\u015f\u0131ndaki t\u00fcm asal say\u0131lar tek say\u0131d\u0131r. Asal say\u0131lar k\u00fcmesi, { 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, … …<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1404,1403],"tags":[7208,7292,7293],"class_list":["post-3094","post","type-post","status-publish","format-standard","hentry","category-matematik-odevleri","category-odevler","tag-asal-sayilar","tag-fermat-teoremi","tag-sayma-sayisi"],"_links":{"self":[{"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/posts\/3094","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/comments?post=3094"}],"version-history":[{"count":0,"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/posts\/3094\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/media?parent=3094"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/categories?post=3094"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/tags?post=3094"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}