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 $\boxed{\textbf{(C)}\ 12}$ .

1973 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 5	Followed by Problem 7
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1973 AHSME Problems/Problem 6

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