Bosco

     2003,  2011,  2017,  2027,  2029       

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In order not to have two yellows next to each other,  

the three yellows must occupy squares 1, 3, and 5. 

That leaves squares 2 and 4 to fill, and two colors  

to do it with. There are only four ways. 

Blue Blue     Red Red     Blue Red     Red Blue  

So, the answer is four ways.  

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     9 • 5 • 3 = 135       Last two digits are 35    

     1^(2^3) = 1⁸ = 1     One raised to any positive power is still one,    

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     (5 x 6 x 7 x 8) + 1  =  1680 + 1  =  1681    

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250 in base 10 is FA in base 16.    

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Disclosure:  Maxematics gave me the idea how to find the answer       

when s/he pointed out that the product would be a multiple of nine,     

and therefore the digits in the product have to add up to a multiple     

of nine.  After that, I used brute force.    

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