P:1793

P:1793 (Ioana Roman)]]

Determinați cel mai mic număr 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 care are loc egalitatea $1+{\overline {abcd}}=88\times {\overline {cd}}$.

Soluție

Egalitatea din enunț se scrie în mod echivalent $1+{\overline {ab}}\times 100+{\overline {cd}}=88\times {\overline {cd}}$, ceea ce conduce la

$${\overline {ab01}}=87\times {\overline {cd}}.$$

Cum doar produsul $7\times 3$ are ultima cifră egală cu $1$, deducem că $d=3$.

Avem produsele

$$87\times 13=1131$$

$$87\times 23=2001$$

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 87 \times 33 = 2871} 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 87 \times 43 = 3741} 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 87 \times 53 = 4611} 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 87 \times 63 = 5481} 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 87 \times 73 = 6351} 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 87 \times 83 = 7221} 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 87 \times 93 = 8091}

de unde deducem că doar produsul 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 87 \times 23 = 2001} este cel care corespunde.

În concluzie, unicul număr 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 care are loc egalitatea 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+\overline{abcd}= 88 \times \overline{cd}} 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 \overline{abcd} = 2023.}