1957 AMC 12 Problem 19

Attempt Problem 19 of the 1957 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 1957 AMC 12 solutions, or check the answer key.

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

19.

The base of the decimal number system is ten, meaning, for example, that

123=1\cdot10^2+2\cdot10+3.

In the binary system, which has base two, the first five positive integers are

1,

10,

11,

100,

101.

The numeral

10011

in the binary system would then be written in the decimal system as:

19

40

10011

11

7

Assign powers

2^4,2^3,\ldots,2^0

to the five binary places

Only the first, fourth, and fifth digits are nonzero

Expanding by binary place value,

\begin{aligned} (10011)_2 &=1\cdot2^4+1\cdot2^1+1\cdot2^0\\ &=16+2+1=19. \end{aligned}

Therefore, the correct answer is A.

Problem 19 in other years