336

84

That would be the smallest number that is evenly divisible by all three. Since 12 is divisible by 3 you don't even need to worry about 3. You need two things: The highest common factor of the two numbers (7 & 12 here) and the product of them (7*12) 7 is a prime number so it's only factors are 1 & 7. That makes this easy, 12 is not divisible by 7, so the highest common factor is 1. Then the product 7*12=84 In any other situation you would divide that number by the highest common factor of the two numbers, since it's 1 that won't make any difference. If it had been 4 & 10 their highest common factor is 2. So you'd multibly 4*10=40 and divide by 2: 40/2=20.

1) The approximate answer is: less than one month. The exact answer is: the question does not correspond to the reality because those planets are about never in the same straight line. 2) The reason for this is that their orbits are usually not in the same plane. "All planetary orbital planes wobble around the invariable plane, meaning that they rotate around its axis while their inclinations to it vary, both of which are caused by the gravitational influence of the other planets. " Source and further information: http://en.wikipedia.org/wiki/Invariable_plane Here some orbital characteristics: "[Earth] Inclination 1°34'43.3" to Invariable plane" "[Earth] Orbital period 365.256366 days 1.0000175 yr" http://en.wikipedia.org/wiki/Earth "[Mercury] Inclination 7.005° 3.38° to Sun’s equator" "[Mercury] Orbital period 87.969 1 d (0.240 846 a)" http://en.wikipedia.org/wiki/Mercury_(planet%29 "[Venus] Inclination 3.394 71° 3.86° to Sun's equator" "[Venus] Orbital period 224.700 69 day 0.615 197 0 yr" http://en.wikipedia.org/wiki/Venus_(planet%29 But let us assume in the following that the planets were in the same plane anyway, so we can find approximate conjunctions. Let us consider a very simple model where the planets move in the same direction with constant velocity on circular, coplanar orbits around the sun. 3) If the three planets had exact orbital periods of 12*30 days, 7*30 days and 3*30 days, and were at a particular time all in a straight line *with the Sun*, they would come back to the same position after exactly 84*30 days (84 "months"). In those 84 "months": - Earth makes 7 turns - Venus makes 12 turns - Mercury makes 28 turns But if we take a circular coordinates system with the Sun as a center and Earth as the origin of angles (relativ to Earth), in those 84 "months": - Venus makes 5 turns (12 -7) - Mercury makes 21 turns (28 - 7) Seen from the Earth: starting from a transit of Mercury (Mercury on the line between Eath and the Sun), the angle between Mercury and the Sun will: - grow to a positive maximum value (where the straight line between Mercury and the Earth is the tangens of Mercury orbit; - decrease to zero (Mercury on the other side of the Sun); - grow to a maximum in absolute value, but this time in the other direction - decrease to zero (back to the transit of Mercury) The idea is that at the time where the angle of Mercury to the sun (viewed from the Earth) is growing and is short before the maximum, the angle of Venus could have the same value and be diminishing. But as Mercury has a much higher velocity on its orbit, it could shortly after this by decreasing reach again the value of the angle of Venus. 4) Experimental confirmation: Here are the dates of the conjunctions of Mercury and Venus in right ascension, as see from the Earth, between 2005 and 2020: "January 14, 2005 00:40:51 Mercury 21' south of Venus 18.5° West March 28, 2005 22:31:53 Mercury 4°49' north of Venus 1.5° West June 27, 2005 20:18:55 Mercury 5' south of Venus 23.3° East July 7, 2005 08:21:22 Mercury 1°38' south of Venus 25.8° East January 17, 2006 02:23:03 Mercury 7°53'south of Venus 6.5° West November 7, 2006 13:36:58 Mercury 1°14' south of Venus 2.8° East August 15, 2007 22:59:23 Mercury 10°04' north of Venus 1.8° East February 26, 2008 02:34:04 Mercury 1°20' north of Venus 26.1° West March 23, 2008 10:13:50 Mercury 1°03' south of Venus 20.3° West June 8, 2008 00:51:42 Mercury 2°59' south of Venus 0.3° West August 23, 2008 05:08:57 Mercury 1°15' south of Venus 20.5° East September 11, 2008 04:50:56 Mercury 3°34' south of Venus 25.4° East March 27, 2009 11:31:44 Mercury 10°37' south of Venus 4° West August 15, 2011 23:17:56 Mercury 6°21' south of Venus 1.3° West June 1, 2012 20:40:22 Mercury 12' north of Venus 6.7° East March 6, 2013 07:21:59 Mercury 5°20' north of Venus 5° West May 25, 2013 03:52:17 Mercury 1°22' north of Venus 15° West June 20, 2013 17:37:02 Mercury 1°57' south of Venus 22° East January 7, 2014 10:28:53 Mercury 6°27' south of Venus 5.9° East October 17, 2014 08:02:46 Mercury 2°44' south of Venus 1.7° West December 30, 2014 07:25:46 Mercury 3°44' south of Venus 12.6° East January 5, 2015 16:30:15 Mercury 1°40' south of Venus 16.1° East August 5, 2015 08:52:16 Mercury 8°11' north of Venus 13.1° East May 13, 2016 20:49:07 Mercury 26' south of Venus 6.5° West July 16, 2016 17:39:09 Mercury 32' north of Venus 10.9° East August 27, 2016 04:57:57 Mercury 5°16' south of Venus 22.1° East March 16, 2017 23:21:32 Mercury 9°33' south of Venus 9.5° East December 15, 2017 16:04:48 Mercury 2°14' north of Venus 5.9° West March 5, 2018 18:28:59 Mercury 1°24' north of Venus 13.4° East March 18, 2018 01:16:29 Mercury 3°53' north of Venus 16.4° East October 14, 2018 15:20:14 Mercury 6°49' north of Venus 15.8° East July 24, 2019 10:32:05 Mercury 5°43' south of Venus 5.8° West September 13, 2019 21:35:21 Mercury 20' south of Venus 8.5° East October 30, 2019 08:29:03 Mercury 2°43' south of Venus 20.3° East May 22, 2020 07:53:55 Mercury 54' south of Venus 18.6° East" Source and further information: http://en.wikipedia.org/wiki/Conjunction_(astronomy_and_astrology%29 At the following dates, there is another conjunction less than a month later: June 27, 2005 February 26, 2008 August 23, 2008 May 25, 2013 December 30, 2014 March 5, 2018 Particularly interesting is December 30, 2014, with the next conjunction 6 days later on January 5, 2015.