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periodic decimal fractions

Source: 1968-69 Germany R4 12.1

October 13, 2024
number theoryalgebra

Problem Statement

Every nonnegative periodic decimal fraction represents a rational number, also in the form pq\frac{p}{q} can be represented (pp and qq are natural numbers and coprime, p≥0p\ge 0, q>0)q > 0). Now let a1a_1, a2a_2, a3a_3 and a4a_4 be digits to represent numbers in the decadic system. Let a1≠a3a_1 \ne a_3 or a2≠a4a_2 \ne a_4.Prove that it for the numbers:
z1=0,a1a2a3a4‾=0,a1a2a3a4a1a2a3a4...z_1 = 0, \overline{a_1a_2a_3a_4} = 0,a_1a_2a_3a_4a_1a_2a_3a_4... z2=0,a4a1a2a3‾z_2 = 0, \overline{a_4a_1a_2a_3} z3=0,a3a4a1a2‾z_3 = 0, \overline{a_3a_4a_1a_2} z4=0,a2a3a4a1‾z_4 = 0, \overline{a_2a_3a_4a_1}
In the above representation p/qp/q always have the same denominator.
[hide=original wording]Jeder nichtnegative periodische Dezimalbruch repr¨asentiert eine rationale Zahl, die auch in der Form p/q dargestellt werden kann (p und q nat¨urliche Zahlen und teilerfremd, p >= 0, q > 0). Nun seien a1, a2, a3 und a4 Ziffern zur Darstellung von Zahlen im dekadischen System. Dabei sei a1 ≠\ne a3 oder a2 ≠\ne a4. Beweisen Sie! Die Zahlen z1 = 0, a1a2a3a4 = 0,a1a2a3a4a1a2a3a4... z2 = 0, a4a1a2a3 z3 = 0, a3a4a1a2 z4 = 0, a2a3a4a1 haben in der obigen Darstellung p/q stets gleiche Nenner.
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