To write the decimal number 37 in binary number system, how many binary digits are required?

Question

Accepted Answer

To determine how many binary digits are required to represent the decimal number 37 in the binary number system, we need to convert the decimal number to binary and count the number of digits in the binary representation.

1. **Conversion of 37 to Binary:**
   - Start by dividing the number by 2 and recording the remainder.
   - Continue dividing the quotient by 2 until the quotient is 0, recording each remainder.
   - The binary representation is obtained by reading the remainders from bottom to top.

Let's perform the conversion:

- 37 ÷ 2 = 18, remainder = 1
   - 18 ÷ 2 = 9, remainder = 0
   - 9 ÷ 2 = 4, remainder = 1
   - 4 ÷ 2 = 2, remainder = 0
   - 2 ÷ 2 = 1, remainder = 0
   - 1 ÷ 2 = 0, remainder = 1

Reading the remainders from bottom to top, the binary representation of 37 is 100101.

2. **Counting the Binary Digits:**
   - The binary number 100101 consists of 6 digits.

Therefore, 6 binary digits are required to represent the decimal number 37 in the binary number system. This supports the given correct answer, Option 3.

**Explanation of Why Other Options are Incorrect:**

- **Option 1 (4 digits):** A binary number with 4 digits can represent numbers from 0 to 15 (2^4 - 1). Since 37 is greater than 15, 4 digits are insufficient.

- **Option 2 (5 digits):** A binary number with 5 digits can represent numbers from 0 to 31 (2^5 - 1). Since 37 is greater than 31, 5 digits are insufficient.

- **Option 4 (7 digits):** While a binary number with 7 digits can indeed represent 37, it is more than necessary. The smallest number of digits required is 6, as determined by the conversion process.

Thus, Option 3 is the correct choice, as it accurately reflects the number of binary digits needed to represent the decimal number 37.