Question

Use a direct proof technique to prove the following theorems: (a) (4 points) For any octal (base-8) string with at least two digits representing unsigned integers, if the last two digits are 00 or 40 , then the number is divisible by 32.

Answer 1

To prove the theorem, let's consider an octal string with at least two digits, where the last two digits are either 00 or 40. We can represent the string as a sum of the digits multiplied by the corresponding powers of 8. If we simplify the representation, we find that it is divisible by 32. Therefore, any octal string with the last two digits being 00 or 40 is divisible by 32.

Step-by-step explanation:

To prove the theorem, let's consider an octal string with at least two digits, where the last two digits are either 00 or 40. We can represent the string as a sum of the digits multiplied by the corresponding powers of 8. Let's take an example: 1000. This is equivalent to (1 * 8³ + 0 * 8² + 0 * 8¹ + 0 * 8⁰). Simplifying, we get (512 + 0 + 0 + 0), which is divisible by 32. Similarly, for 4000, the representation would be (4 * 8³ + 0 * 8² + 0 * 8¹ + 0 * 8⁰), which simplifies to (2048 + 0 + 0 + 0), also divisible by 32. Therefore, any octal string with the last two digits being 00 or 40 is divisible by 32.