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You already found that the slope of the line is -3/5. We can use one of the points and the slope to complete the equation in Ax + By = C form.

Let's use the point (-4, 2).

Slope-Intercept Form: We know the slope (m) is -3/5 and the point is (-4, 2) represented as (x1, y1). We can use the slope-intercept form to find the y-intercept (b):

y = mx + b y1 = m * x1 + b

Plugging in the values:

2 = (-3/5) * (-4) + b 2 = 12/5 + b

Solving for b:

b = 2 - 12/5 b = -2/5

Therefore, the equation in slope-intercept form is:

y = (-3/5)x - 2/5

Converting to Ax + By = C: To get the equation in Ax + By = C form, we need to manipulate the slope-intercept form:

y = (-3/5)x - 2/5 5y = -3x - 2 (multiply both sides by 5 to eliminate the fraction)

Now, the equation is in the form Ax + By = C, where A = -3, B = 5, and C = -2.

Greatest Common Divisor (GCD): The greatest common divisor (GCD) of -3, 5, and -2 is 1. However, A needs to be positive according to the prompt.

Making A positive: We can multiply the entire equation by -1 to make A positive:

(-1) * (5y) = (-1) * (-3x) + (-1) * (-2)

5y = 3x + 2

3x - 5y = -2

Therefore, the standard form of the line is 3x - 5y = -2.

Since angle EDF is inscribed in a circle, we know that the measure of the central angle subtended by arc DF is double the measure of angle EDF. Therefore, the central angle for arc DF is 29 x 2 = 58 degrees.

Since the circumference of the circle is 24 units and the central angle for arc DF is 58 degrees, we can use the formula for the circumference of a circle to find the radius of circle G.

C = 2πr
24 = 2πr
r = 12/π

Now, we can use the formula for the length of an arc, given by:

Length of arc = (central angle/360) * circumference

For arc DE, the central angle is 360 - 132 - 58 = 170 degrees.

Length of arc DE = (170/360) * 2π(12/π)

Length of arc DE = (17/36) * 24

Length of arc DE = 8/3 units

Therefore, the length of arc DE is 8/3 units.

We can analyze this problem by considering the divisibility rules for 2, 3, 4, 5, 6, 8, and 9:

Divisibility Rules:

A number is divisible by 2 if the last digit is even (0, 2, 4, 6, or 8).

A number is divisible by 3 if the sum of its digits is divisible by 3. (Given in the problem statement)

A number is divisible by 4 if the last two digits are divisible by 4. (Not applicable here since the number ends in 0)

A number is divisible by 5 if the last digit is a 0 or a 5.

A number is divisible by 6 if it's divisible by both 2 and 3.

A number is divisible by 8 if the last three digits are divisible by 8. (Not applicable here since the number ends in 0)

A number is divisible by 9 if the sum of its digits is divisible by 9. (This is similar to the divisibility rule for 3, but a stronger condition)

Analysis:

Ending in 0: Since the number ends in 0, it is automatically divisible by 2 and 5 (based on the divisibility rules).

Sum of Digits Divisible by 3: The problem states that the sum of the digits is divisible by 3. This means the number itself must also be divisible by 3 (based on the divisibility rule for 3).

Divisibility by Other Numbers:

Looking at the remaining numbers (4, 6, 8, and 9), we can see that:

4: Not necessarily divisible. Just because the last digit is 0 (divisible by 2), it doesn't guarantee divisibility by 4 (requires the last two digits to be divisible by 4).

6: Since the number is divisible by both 2 (ending in 0) and 3 (given in the problem statement), it is automatically divisible by 6.

8: Not necessarily divisible. Similar to 4, just because the number ends in 0 (divisible by 2), it doesn't guarantee divisibility by 8 (requires the last three digits to be divisible by 8).

9: Not necessarily divisible. The sum of digits being divisible by 3 is a weaker condition than being divisible by 9. So, a number divisible by 3 might not necessarily be divisible by 9.

Conclusion:

Therefore, a number ending in 0 with a sum of digits divisible by 3 is necessarily divisible by 2, 3, and 6 (out of the given numbers). There are 3​ such divisors.

We can solve this problem using the divisibility rule for 9. This rule states that a number is divisible by 9 if the sum of its digits is divisible by 9.

Divisibility Rule for 9: A number (like 8*10^18 + 1^18) is divisible by 9 if the sum of its digits is divisible by 9.

Simplifying the Expression:

8*10^18 can be written as 800,000,000,000,000,000 (eight hundred quadrillion).

The sum of the digits in 800,000,000,000,000,000 is 8 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 8 = 16.

Remainder When Divided by 9:

We need to find the remainder when 8*10^18 + 1^18 (which has a digit sum of 16) is divided by 9.

Since 18 is a multiple of 2, 1^18 will always be 1. So, we can focus on the divisibility of 800,000,000,000,000,000 (with a digit sum of 16).

We can find the remainder by subtracting the nearest multiple of 9 that is less than 16: 16 - 9 (nearest multiple of 9 less than 16) = 7.

Therefore, the remainder when 8*10^18 + 1^18 is divided by 9 is 7.

The answer is 456.