Question

Design an algorithm for finding all the factors of a positive integer. For example, in the case of the integer 12 , your algorithm should report the values \(1,2,3,4,6\), and 12 .

Short answer

Loop through numbers from 1 to `n`, check for divisibility, and collect all divisors.

Step-by-step solution

Understand the Problem

You need to find all numbers that divide a given positive integer without leaving a remainder. For instance, for the number 12, you should find all integers from 1 to 12 that can divide 12 exactly.

Initialize a Loop

Start a loop where a variable, let’s call it `i`, iterates from 1 up to and including the integer `n`, for which we want to find the factors.

Check for Divisibility

For each iteration, check if the integer `n` is divisible by the current loop variable `i` using the modulus operation. If `n % i == 0`, then `i` is a factor of `n`.

Collect Factors

If `i` divides `n` perfectly (i.e., if the remainder is 0), then add `i` to the list of factors. Continue this process until the loop reaches `n`.

Conclusion

After the loop ends, the list will contain all the factors of the integer `n`. For instance, for `n = 12`, the list of factors will be [1, 2, 3, 4, 6, 12].

Key concepts

Factors of an Integer

When we talk about factors of a positive integer, we're discussing the numbers that can be multiplied together to get that integer. For instance, if we take the number 12, its factors are 1, 2, 3, 4, 6, and 12, because each of these numbers divides 12 exactly without leaving any remainder.

Understanding factors is crucial for several math concepts, including simplifying fractions and solving algebraic equations. To find the factors of a number:

• Start with the number 1, which is a universal factor since it divides any number exactly once.
• Continue testing each successive integer to see if it divides the target number without leaving a remainder.
• Once you reach the number itself, you've got the complete list of factors.

Factors help us understand the divisibility of numbers better and are foundational in number theory and its applications.

Divisibility

Divisibility refers to the ability of one number to be divided by another without leaving a remainder. It's a fundamental concept much like factors but viewed from a slightly different perspective.

To determine the divisibility of a number, like finding out if 3 is a factor of 12, you check if dividing one number by another leaves no remainder:

• If a number divides another without leaving a remainder, it's termed 'divisible'. For example, 12 divided by 3 equals 4, with no remainder, so 12 is divisible by 3.
• Various divisibility rules can quickly help you identify if one number divides into another, such as checking if a number ends with an even digit to determine divisibility by 2.

The concept of divisibility is essential in mathematics, especially in simplifying principal mathematical operations like fractions and finding least common denominators.

Modulus Operation

The modulus operation is a crucial tool for determining the remainder when one integer is divided by another. It helps in several algorithm designs, especially when factors and divisibility are concerned.

Performed using the % operator in most programming languages, the modulus operation expresses the remainder of a division:

• For example, if we use 12 % 5, the result is 2 because 5 goes into 12 twice fully, and the remainder is 2.
• In the context of factors, checking if an integer `n` is divisible by another integer `i` by evaluating `n % i == 0` is common. If true, `i` is indeed a factor of `n`.

Understanding and applying the modulus operation allows learners to efficiently design algorithms that determine factors and establish divisibility — essential skills in programming and problem-solving.