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1.) In a solar system of $$n$$ planets, Zorn the World Conqueror can invade $$m$$ planets at a time, but once there are less than $$m$$ free worlds left, he stops. If he invades $$13$$ at a time then there are $$6$$ left, and if he invades $$14$$ at a time then there are $$5$$ left. If this solar system has more than $$100$$ planets, what is the smallest number of planets it could have?

2.) A palindrome is a number that reads the same forward and backward. The largest possible 4-digit palindrome in base 3 can be expressed as a 3-digit palindrome in another base $$b>2$$. What is $$b$$?

Dec 18, 2018

#1
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$$\text{let }N \text{ be the number of planets in the solar system}\\ N \pmod{13} = 6\\ N \pmod{14}=5 \\ \text{13 and 14 have no common factors so any solutions will be of the form }\\ x= r + 13\cdot 14 k = r+182k,~r,k \in \mathbb{N}\\ \text{we can eyeball that 19 is a solution and so all solutions will be }\\ N = 19 + 182k \text{, and the first of these such that N > 100 is }\\ N = 19+182 = 201$$

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Dec 18, 2018
#2
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$$\text{well the largest 4 base 3 digit palindrome is }\\ p = 2222 = 80d\\ \text{I don't see anything more clever than listing this in a few bases until we find a palindrome}\\ \text{In order to have 3 digits it must be that }b^2 < 80d < b^3 \text{ so }\\ 5 \leq b \leq 8\\ \left( \begin{array}{cc} 5 & \{3,1,0\} \\ 6 & \{2,1,2\} \\ 7 & \{1,4,3\} \\ 8 & \{1,2,0\} \\ \end{array} \right)\\ \text{and we see that base 6 is the only base that produces a 3 digit palindrome of }80d$$

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Dec 18, 2018