Question 1174804
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Edit: I've included the year 2018 as part of the answer after rereading the problem.


The Fields Medal is awarded every 4 years.
The Birkhoff Prize is awarded every 3 years.


With 2010 as a reference point for the Fields Medal, let's write a few numbers that are 4 spaces ahead and behind this reference year.
{..., 1990, 1994, 1998, 2002, <font color=red>2006</font>, 2010, 2014, <font color=blue>2018</font>, 2022, 2026, 2030, ...}


Do a similar thing for the Birkhoff Prize years. We start at 2009. Add and subtract multiples of 3 to represent possible years the prize is given
{..., 1994, 1997, 2000, 2003, <font color=red>2006</font>, 2009, 2012, 2015, <font color=blue>2018</font>, 2021, 2024...}


The year marked in red is where the two prizes line up, and also the smallest such year in the 21st century (which starts at the year 2000). 


Note how 4*3 = 12 is the LCM of 4 and 3. This tells us that the two awards will line up every 12 years. So 2006+12 = <font color=blue>2018</font> is the next time the two awards will be given together. To help confirm this, the year 2018 is found in both sets listed above. 


Also, note that 2006-12 = 1994 was the last time prior to the 2006 year, is when the two awards lined up. The year 1994 is found in both sets listed above.


These years are when the two awards lined up
{..., 1994, <font color=red>2006</font>, <font color=blue>2018</font>, ...}


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Answer: The year <font color=red>2006</font> and <font color=blue>2018</font>
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