# 2007 iTest Problems/Problem 45

## Problem

Find the sum of all positive integers $B$ such that $(111)_B=(aabbcc)_6$, where $a,b,c$ represent distinct base $6$ digits, $a\neq 0$.

## Solution

Using the definition of base numbers, the equation can be rewritten as $$B^2 + B + 1 = 9072a + 252b + 7c$$ $$\frac{B^2 + B + 1}{7} = 1296a + 36b + c$$

To find the values of $B$, use casework for values of $a$ since $a$ has the most influence on the value of $7(1296a + 36b + c) - 1$. Casework will be heavy, but a few tips can lighten the load. First, since $B^2 + B$ is one less than a multiple of $7$, $B$ is congruent to $2$ or $4$ modulo $7$. Second, once $\frac{B^2 + B + 1}{7} > 1296a + 185$, $B$ can not be higher for a given $a$. Third, use estimation to approximate the lower bound for a given $a$.

• If $a = 1$, then $B^2 + B$ is just more than $9072$. The first few values of $B$ that work are $95$, $100$, $102$, and $107$. Testing each case, $B = 100$ when $b = 4$ and $c = 3$.
• If $a = 2$, then $B^2 + B$ is just more than $18144$. The first few values of $B$ that work are $137$, $142$, $144$. Testing each case, $B = 137$ when $b = 3$ and $c = 1$.
• If $a = 3$, then $B^2 + B$ is just more than $27216$. The first few values of $B$ that work are $165$ and $170$. After testing each case, no values of $B$ work when $a = 3$.
• If $a = 4$, then $B^2 + B$ is just more than $36288$. The first few values of $B$ that work are $191$ and $193$. After testing each case, once again, no values of $B$ work when $a = 4$.
• If $a = 5$, then $B^2 + B$ is just more than $45360$. The first few values of $B$ that work are $214$ and $219$. After testing each case, yet again, no values of $B$ work when $a = 5$.

In summary, the only possible values of $B$ are $100$ and $137$, and the sum of the values equals $\boxed{237}$.

## See Also

 2007 iTest (Problems) Preceded by:Problem 44 Followed by:Problem 46 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 • 51 • 52 • 53 • 54 • 55 • 56 • 57 • 58 • 59 • 60 • TB1 • TB2 • TB3 • TB4
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