15678: Difference between revisions
Paul Matei (talk | contribs) Pagină nouă: '''14441 (Constantin Apostol, Rm. Sărat)''' ''Arătați că oricare ar fi cifrele '' <math>a</math> '' și <math>b</math>, numărul <math>\overline{ab15}</math> are un număr par de divizori naturali.'' '''Soluție:''' ''Se știe că un număr natural cu un număr impar de divizori este pătrat perfect.De asemenea, un pătrat perfect care are cifra unităților 5 are cifra zecilor 2.Cu acestea, dacă presupunem că numărul <math>\overline{ab15}</math> are un număr i... |
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''' | '''15678 (Cristina Vijdeluc și Mihai Vijdeluc)''' | ||
'' | ''Aflați toate numerele de forma ''<math>\overline{abcd}</math> ''pentru care<math display="block">\overline{abcd} = 2021+5(a-c+b-d+1).</math> | ||
'''Soluție:''' | '''Soluție:''' | ||
Ultima cifră numărului <math>5(a-c+b-d+1)</math> poate fi <math>0</math> sau <math>5</math>. Atunci ultima cifră a numărului <math>2021+5(a-c+b-d+1)</math> poate fi <math>1</math> sau <math>6</math>, de unde deducem că <math>d=1</math> sau <math>d=6</math>. | |||
'' | '''1.''' Pentru <math>d=1</math> relația devine <math>\overline{abcd} = 2021+5(a-c+b)</math>. Deoarece <math>a-c+b \le 18</math> avem <math>2021+5(a-c+b) \le 2111</math> și, cum <math>\overline{abcd}= 2021+5(a-c+b)</math>, obținem <math>a \le 2</math>. | ||
Pentru <math>a=2</math> relația dată devine <math>\overline{2bc1} = 2021+5(2-c+b)</math> sau <math>2001+100b+10c=2031-5c+5b</math>, de unde <math>95b+15c=30</math>. | |||
De aici <math>b=0</math> și <math>c=2</math>. Numărul căutat este <math>2021</math>. | |||
Pentru <math>a=1</math> nu obținem nicio soluție. | |||
'''2.''' Pentru <math>d=6</math> relația dată devine <math>\overline{abc6} = 2021+5(a-c+b-5)</math>. Deoarece <math>a-c+b-5 \le 13</math> avem <math>2021+5(a-c+b) \le 2086</math> și cum <math>\overline{abc6} = 2021+5(a-c+b-5)</math>, obținem <math>a \le 2</math>. | |||
Pentru <math>a=2</math> relația dată devine <math>\overline{2bc6} = 2021+5(2-c+b-5)</math> sau <math>2006+100b+10c=2006-5c+5b</math>, de unde <math>95b+15c=0</math>. De aici <math>b=0</math> și <math>c=0</math>. | |||
Numărul căutat este <math>2006</math>. | |||
Pentru <math>a=1</math> nu obținem nicio soluție. | |||
Latest revision as of 08:40, 1 December 2024
15678 (Cristina Vijdeluc și Mihai Vijdeluc)
Aflați toate numerele de forma Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{abcd}} pentru careFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{abcd} = 2021+5(a-c+b-d+1).}
Soluție: Ultima cifră numărului Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 5(a-c+b-d+1)} poate fi Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0} sau . Atunci ultima cifră a numărului Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2021+5(a-c+b-d+1)} poate fi Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1} sau Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 6} , de unde deducem că Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d=1} sau Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d=6} .
1. Pentru Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d=1} relația devine Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{abcd} = 2021+5(a-c+b)} . Deoarece Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a-c+b \le 18} avem Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2021+5(a-c+b) \le 2111} și, cum Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{abcd}= 2021+5(a-c+b)} , obținem Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \le 2} .
Pentru Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a=2} relația dată devine Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{2bc1} = 2021+5(2-c+b)} sau Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2001+100b+10c=2031-5c+5b} , de unde Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 95b+15c=30} .
De aici Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=0} și Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c=2} . Numărul căutat este Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2021} .
Pentru Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a=1} nu obținem nicio soluție.
2. Pentru Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d=6} relația dată devine Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{abc6} = 2021+5(a-c+b-5)} . Deoarece Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a-c+b-5 \le 13} avem Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2021+5(a-c+b) \le 2086} și cum Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{abc6} = 2021+5(a-c+b-5)} , obținem Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \le 2} .
Pentru Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a=2} relația dată devine Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{2bc6} = 2021+5(2-c+b-5)} sau Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2006+100b+10c=2006-5c+5b} , de unde Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 95b+15c=0} . De aici Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=0} și Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c=0} . Numărul căutat este Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2006} .
Pentru Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a=1} nu obținem nicio soluție.