Question

The Gregorian epact is the number of days between January \(1^{\text {st}}\) and the previous new moon. This value is used to figure out the date of Easter. It is calculated by these formulas (using int arithmetic): \\[ C=y e a r / / 100 \\] \\[ e p a c t=(8+(C / / 4)-C+((8 C+13) / / 25)+11(\text { year } \% 19)) \% 30 \\] Write a program that prompts the user for a 4-digit year and then outputs the value of the epact.

Short answer

Calculate century C, then find the epact with the given formula steps, and output it.

Step-by-step solution

Input the Year

Prompt the user to enter a 4-digit year. Ensure that the input is taken as an integer since we will be performing arithmetic operations on it.

Calculate the Century (C)

Calculate the century by dividing the input year by 100 using integer division. This is done with the expression: \[ C = \text{{year}} // 100 \]

Calculate Intermediate Value (A)

Compute a temporary value involved in the formula which combines several divisions and multiplications. The specific term is \[ A = \left(\left(8 \times C + 13\right) // 25\right) \].

Calculate the Intermediate Step

Calculate the intermediate value needed for the epact, integrating parts of the overall formula:\[ \text{{intermediate}} = \left(8 + (C // 4) - C + A + 11 \times (\text{{year}} \, \% \, 19)\right) \]

Compute the Epact

Compute the final value for the epact by taking the intermediate result mod 30:\[ \text{{epact}} = \text{{intermediate}} \% 30 \]

Output the Result

Print the calculated epact value as the result to the user, indicating this is the epact for the input year.

Key concepts

Integer Division

Integer division is a fundamental concept in Python programming. It is used to divide one number by another and then truncate the result to an integer. In Python, the operator for integer division is `//`. For instance, if you divide 10 by 3 using integer division, it results in 3 because it cuts off the decimal portion and just leaves the whole number.
The exercise uses integer division to calculate the century component from a year. When you carry out `year // 100`, it effectively removes the last two digits, providing the century number. For example, the year 2022 divided by 100 using integer division yields 20. This is a crucial step in the calculation process of the epact.

• Use `//` to perform integer division.
• Integer division discards the decimal.
• Essential for operations where whole numbers are needed.

Modulo Operation

The modulo operation is another essential arithmetic operation in Python. It provides the remainder left after division. The operator for modulo is `%`. For example, `10 % 3` results in 1 because when 10 is divided by 3, the remainder is 1.
In the context of the given exercise, the modulo operation `% 19` is used to find the remnant of the year when divided by 19 and to compute the final epact value by taking the result modulo 30. These operations help adjust the calendar year to match lunar phases.

• Helps in obtaining the remainder after a division.
• Used in the epact formula to adjust calculations.
• Facilitates periodic calculations like finding days of the week.

Arithmetic Operations

Arithmetic operations involve basic calculations such as addition, subtraction, multiplication, and division on numbers. In Python, they are essential for performing mathematical tasks. The epact calculation relies heavily on arithmetic operations.
To compute the epact, a series of arithmetic operations combine integer division and the modulo operation as well as additions and subtractions. For instance, expressions like `(8 + (C // 4) - C + A)` are evaluated to assess different components of the formula. Understanding these operations is crucial to parse and implement the formula accurately.

• Addition is done with `+`.
• Subtraction uses `-`.
• Multiplication is symbolized by `*`.
• Combination of operations for complex calculations.

User Input

Capturing user input in Python is straightforward using the `input()` function. It allows the program to receive data from the user. Since the program is tasked with determining the epact, it prompts the user to enter a 4-digit year.
The `input()` function by default returns the data as a string, so it must be converted to an integer using `int()`, especially since arithmetic operations will be performed. This conversion is a critical step in ensuring that the data can be used in numerical calculations.

• Use `input()` to get data from the user.
• Convert input to appropriate data types like `int()` when necessary.
• Vital for interactive programming tasks.

Conditional Statements

Though not explicitly mentioned in the original exercise, conditional statements are an integral part of programming, used to make decisions in a program flow. In Python, conditional statements include `if`, `else`, and `elif`.
While the exercise here focuses on arithmetic computation, in a full-featured program, you might use conditional statements to check if the user's input is valid or if the computed epact meets particular criteria. For example, you may want to check if the year is indeed a 4-digit number.

• `if` statements check conditions.
• Use `else` to define alternative actions.
• `elif` serves as "else if" for multiple conditions.