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Printjmc
number theory intermediate
Problem
If the digit represented by satisfies the following addition problem \begin{array}{c@{}c@{\;}c@{}c@{}c@{}c} & & 3 & 2 & 1 & \triangle_6\\ & & & \triangle & 4 & 0_6\\ &+ & & & \triangle & 2_6\\ \cline{2-6} & & 4 & 2 & \triangle & 1_6\\ \end{array}what is the value of ?
Solution
We notice that in the units column, it's impossible for . So, it must be the case that . That would mean . We can check that our answer works by plugging our value for triangle into the original problem: \begin{array}{c@{}c@{\;}c@{}c@{}c@{}c} & & 3 & 2 & 1 & 5_6\\ & & & 5 & 4 & 0_6\\ &+ & & & 5 & 2_6\\ \cline{2-6} & & 4 & 2 & 5 & 1_6.\\ \end{array}The addition problem still works, so our answer is correct.
Final answer
5