Track / Stage 6 / 267 of 400 #1267 of 2000

Problem 1267

National olympiad, first round

Number theory Difficulty 6.5 Find the answer

The base of the decimal number system is ten, meaning, for example, that $123 \equal{} 1\cdot 10^2 \plus{} 2\cdot 10 \plus{} 3$ . 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:

$ \textbf{(A)}\ 19 \qquad
\textbf{(B)}\ 40\qquad
\textbf{(C)}\ 10011\qquad
\textbf{(D)}\ 11\qquad
\textbf{(E)}\ 7$

Your answer

Official solution

To convert the binary number $10011_2$ to its decimal equivalent, we need to express it as a sum of powers of 2. Each digit in the binary number represents a power of 2, starting from $2^0$ on the right.

1. Write down the binary number and its corresponding powers of 2:
$10011_2 = 1 \cdot 2^4 + 0 \cdot 2^3 + 0 \cdot 2^2 + 1 \cdot 2^1 + 1 \cdot 2^0$

2. Calculate each term:
$1 \cdot 2^4 = 1 \cdot 16 = 16$
$0 \cdot 2^3 = 0 \cdot 8 = 0$
$0 \cdot 2^2 = 0 \cdot 4 = 0$
$1 \cdot 2^1 = 1 \cdot 2 = 2$
$1 \cdot 2^0 = 1 \cdot 1 = 1$

3. Sum all the terms:
$16 + 0 + 0 + 2 + 1 = 19$

Therefore, the decimal equivalent of the binary number $10011_2$ is $19$ .

The final answer is $\boxed{19}$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.