1973 AMC 12 Problem 6

Attempt Problem 6 of the 1973 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1973 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

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

6

8

12

14

16

Translate

24_b

and

554_b

into base-ten polynomial expressions

After solving the resulting quadratic, enforce that a base containing digit

5

must exceed

5

In base ten,

\begin{aligned} 24_b&=2b+4,\\ 554_b&=5b^2+5b+4. \end{aligned}

Therefore

\begin{aligned} (2b+4)^2&=5b^2+5b+4,\\ b^2-11b-12&=0,\\ (b-12)(b+1)&=0. \end{aligned}

A base is positive and must exceed

5,

b=12.

Therefore, the correct answer is C.

Problem 6 in other years