Question: In the groups Z/nZ introduced above, the element 1 is primitive, so these groups are cyclic. Indeed, each element is expressible as a sum all of whose terms are 1. Any cyclic group with n elements is isomorphic to this group. A second example for cyclic groups is the group of n-th complex roots of unity, given by complex numbers z satisfying zn = 1. These numbers can be visualized as the vertices on a regular n-gon, as shown in blue at the right for n = 6. The group operation is multiplication of complex numbers. In the picture, multiplying with z corresponds to a counter-clockwise rotation by 60°. Using some field theory, the group Fp× can be shown to be cyclic: for example, if p = 5, 3 is a generator since 31 = 3, 32 = 9 ≡ 4, 33 ≡ 2, and 34 ≡ 1.
Try to answer this question if possible: A cyclic group without n elements is considered what?
Answer: unanswerable
Question: In conjunction with the system of months there is a system of weeks. A physical or electronic calendar provides conversion from a given date to the weekday, and shows multiple dates for a given weekday and month. Calculating the day of the week is not very simple, because of the irregularities in the Gregorian system. When the Gregorian calendar was adopted by each country, the weekly cycle continued uninterrupted. For example, in the case of the few countries that adopted the reformed calendar on the date proposed by Gregory XIII for the calendar's adoption, Friday, 15 October 1582, the preceding date was Thursday, 4 October 1582 (Julian calendar).
Try to answer this question if possible: What was the previous day's date by the Julian calendar?
Answer: Thursday, 4 October 1582
Question: PlayStation Network is the unified online multiplayer gaming and digital media delivery service provided by Sony Computer Entertainment for PlayStation 3 and PlayStation Portable, announced during the 2006 PlayStation Business Briefing meeting in Tokyo. The service is always connected, free, and includes multiplayer support. The network enables online gaming, the PlayStation Store, PlayStation Home and other services. PlayStation Network uses real currency and PlayStation Network Cards as seen with the PlayStation Store and PlayStation Home.
Try to answer this question if possible: In what city did Sony hold their 2006 PlayStation Business Briefing?
Answer: Tokyo
Question: The dollar was first based on the value and look of the Spanish dollar, used widely in Spanish America from the 16th to the 19th centuries. The first dollar coins issued by the United States Mint (founded 1792) were similar in size and composition to the Spanish dollar, minted in Mexico and Peru. The Spanish, U.S. silver dollars, and later, Mexican silver pesos circulated side by side in the United States, and the Spanish dollar and Mexican peso remained legal tender until the Coinage Act of 1857. The coinage of various English colonies also circulated. The lion dollar was popular in the Dutch New Netherland Colony (New York), but the lion dollar also circulated throughout the English colonies during the 17th century and early 18th century. Examples circulating in the colonies were usually worn so that the design was not fully distinguishable, thus they were sometimes referred to as "dog dollars".
Try to answer this question if possible: Outside of Mexico, where else was the Spanish dollar minted?
Answer:
Peru