1973 AHSME Problems/Problem 6

Problem

If 554 is the base $b$ representation of the square of the number whose base $b$ representation is 24, then $b$, when written in base 10, equals

$\textbf{(A)}\ 6\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 14\qquad\textbf{(E)}\ 16$

Solution

Write out the numbers using the definition of base numbers. \[554_b = 5b^2 + 5b + 4\] \[24_b = 2b+4\] Since $554_b = (24_b)^2$, we can write an equation. \[5b^2 + 5b + 4 = (2b+4)^2\] \[5b^2 + 5b + 4 = 4b^2 + 16b + 16\] \[b^2 - 11b - 12 = 0\] \[(b-12)(b+1) = 0\] Since base numbers must be positive, $b$ in base 10 equals $\boxed{\textbf{(C)}\ 12}$.

See Also

1973 AHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 5
Followed by
Problem 7
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