Each of the small letters in bold represents a different base x digit from 0 to x-1 to satisfy this alphametic equation. None of the numbers can contain any leading zero.

(x-1)***number** = **integer**

Determine the minimum positive integer value of x such that the above equation has at least one solution. What is the next smallest value of x having this property?

Jer noted "If r is not zero, r+r has a carry so for e+e to end in r, r must be odd." When this is true then the base x is of the form 4y+2.

Then starting with y=4, the following parametric solution finds a solution for every base 18,22,26,30,....

B=y-1, E=y, G=1, I=4y-1, M=4y+1, N=4y, R=2y+1, T=2, U=4y-2

4y-1 4y 2 y 1 y 2y+1

+ 4y 4y-2 4y+1 y-1 y 2y+1

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4y 4y-2 4y+1 y-1 y 2y+1 0