Working in base 8, state a divisibility criterion for each of the following numbers: 2,3,4,7 and 10 in base 8. Justify each of these rules. Don't forget to leave traces of your construction of each of the rules

To determine the divisibility criterion for a number in base 8, we need to consider the sum of the digits in the number.

1. Divisibility by 2:
For a number to be divisible by 2 in base 8, its last digit must be an even number (i.e., 0, 2, 4, or 6). This can be justified because any even number multiplied by 2 will result in an even number, which means the last digit remains unchanged.

2. Divisibility by 3:
To determine if a number is divisible by 3 in base 8, we need to calculate the sum of its digits. If the sum is divisible by 3, then the number itself is divisible by 3. This can be justified because the divisibility rule for 3 in base 8 is the same as in base 10. The sum of the digits is the same regardless of the base.

3. Divisibility by 4:
To determine if a number is divisible by 4 in base 8, we need to examine its last two digits. If the last two digits form a number that is divisible by 4, then the entire number is divisible by 4. This can be justified because any number multiplied by 4 will result in a number that ends with two zeroes, which means the last two digits will remain unchanged.

4. Divisibility by 7:
Finding a divisibility criterion for 7 in base 8 is a bit more complicated. One way to approach it is to use the method of long division. Divide the number by 7, keeping only the remainder. Continue dividing the remainder by 7 until you reach zero. If at any point you get a remainder of 0, then the number is divisible by 7. Otherwise, it is not divisible.

5. Divisibility by 10:
In base 8, a number is divisible by 10 if it ends with a zero. This can be justified because any number multiplied by 8 (the base) will result in a number that ends with a zero.

In summary:

- Divisibility by 2: The last digit must be an even number.
- Divisibility by 3: The sum of the digits must be divisible by 3.
- Divisibility by 4: The last two digits must form a number divisible by 4.
- Divisibility by 7: Use the method of long division, dividing by 7 until reaching zero, checking the remainders.
- Divisibility by 10: The number must end with a zero.

Answer: The divisibility criterion for each of the numbers in base 8 are as follows:
- Divisible by 2: The last digit is even.
- Divisible by 3: The sum of the digits is divisible by 3.
- Divisible by 4: The last two digits form a number divisible by 4.
- Divisible by 7: The long division method yields no remainder.
- Divisible by 10: The number ends with a zero.