Base Equation

Number Theory Level 4

$\overline{BBB}_{10}=1110_B$ , where $B$ is an integer. Find the average of the sum of all values of $B$ .

Details and Clarifications

If $B=7$ , then $\overline{BBB}$ would equal 777.
Also, if $B=12$ , (or anything for $B>9$ ) $\overline{BBB}=12+12*10+12*100$
Bases can be negative. If confused, read negative integer number base .

The answer is -0.3333.

1 solution

A P
May 30, 2016

$BBB_{10}$ would just be equal to $BBB$ , and $1110_B$ would be equal to $B^3+B^2+B^1$ . First off, $B=0$ is clearly an option. $BBB=B*111$ , so dividing both sides by $B$ would yield $B^2+B+1=111$ . Solving this equation, we get that $-11$ and $10$ are solutions, and averaging these three answers we get $-\frac{1}{3}$ .

Shouldn't it be -0.5?

Brandon Huang - 1 year, 11 months ago

Base Equation

The answer is -0.3333.

1 solution

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