Quandaries
and Queries 

Question: Hi my name is Christine and i would like to know the answer to the following problem: The Parliament of the land of Achronia consists of two houses. The parliament was elected in 2003 for a period of six years beginning on Thursday, the 1st of January 2004, when the two houses had their first sessions. According to the rules, the meetings of the first house must occur every twelve days for the duration of the term, and the meetings of the second house must occur every eighteen days. For example, the second meetings of the first and the second houses were held on the 13th and 19th of January respectively. A new law can be passed on any day when both houses meet, except on a Thursday. On how many days can the parliament members pass new laws during this six year term? It would be very greatly appreciated if you could guide me through it and to get the answer. Thank you very much. 

Hi Christine, The first house meets after 12 days, after 24 days, after 36 days,... That is after periods that are multiples of 12 days. The second house meets after 18 days, after 36 days, after 54 days,... That is after periods that are multiples of 18 days. The first time they meet after January 1, 2004 is after 36 days, since 36 is the smallest number that is a multiple of both 12 and 18. (36 is the least common multiple of 12 and 18.) They meet again after another 36 days, and then again after another 36 days... In the 6 year term of the Parliament there are 2 leap years, 2004 and 2008 so there are 6 365 + 2 = 2192 days in the term. Also ^{2192}/_{36} = 60.89 and hence there are 60 times when both houses meet on the same day. How many of these meetings are on a Thursday? The first one is, January 1, 2004 is a Thursday. The next Thursday is 7 days later and then the next is 7 days after that... Thus the number of days before a meeting of both houses on a Thursday is the smallest number that is a multiple of both 36 and 7. Can you complete the problem now? 

