Question

Să se determine cifrele a, b, c, d, care verifică egalitatea:

$\it\dfrac{\overline{abc}}{10000} = \dfrac{1}{a+b+c+d}$


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Answer 1

$\dfrac{\overline{abc}}{10000} = \dfrac{1}{a+b+c+d} \\ \\ \overline{abc}\cdot (a+b+c+d) = 10000 \\ \\ a+b+c+d = \dfrac{10000}{\overline{abc}}\\ \\ a+b+c+d \leq 9+9+9+9 \leq 36 \\ \\ \overline{abc}~|~10000,\text{ dar }a+b+c+d}<\overline{abc}\\ \\ a+b+c+d = \dfrac{(a+b+c+d)\cdot \overline{abc}}{\overline{abc}}\\ \\ (a+b+c+d)\cdot \overline{abc} = 10000 \\ \\ 10000 = 2^4\cdot 5^4 =2^4\cdot 625 = (2^2\cdot 5)\cdot 500 \\ \\ \text{Sunt singurele posibilitati, deoarece factorul din stanga trebuie sa fie }< 36}\\ \text{iar cel din dreapta}<1000$

$(1)~\overline{abc} = 625 \Rightarrow a=6,b=2,c=5 \\ a+b+c+d = 2^4 = 16 \Rightarrow 6+2+5+d = 16 \Rightarrow d = 3 \\ \\ \Rightarrow a = 6,b = 2,c= 5,d = 3 \\ \\ (2)~\overline{abc} = 500 \Rightarrow a = 5,b=0,c=0 \\ a+b+c+d = 20 \Rightarrow 5+d = 20 \Rightarrow d= 15 ~(F) \\ \\ \Rightarrow (a,b,c,d) = (6,2,5,3)$