# Problem Statement

Find the primes in the first n positive integers.

# Problem Domain Analysis

A prime number is a positive integer that is exactly divisible only by 1 and itself. The first few primes are:

``````2,3,5,7,11,13,17,19,23,29,31,37
``````

All primes apart from 2 are odd.

# Solution Domain Analysis

## Example 1

Let's check if 13 is a prime. If the number we are testing is prime it will have no exact divisors other than 1 and itself. To determine if 13 is prime, we need to divide it in turn by the set of numbers

``````2,3,4,5,.....12
``````

If any of these numbers divide into 13 without remainder, we will know it cannot be prime. This requires 11 calls to mod function. This is very inefficient. Let's look at a different approach.

## Example 2

Let's look at an example problem. If n = 37, we don't need to test all of the numbers 3 through 36 to determine whether n is prime. Instead, we can test only the numbers between 2 and sqrt(37), rounded up. sqrt(37) = 6.08 - we'll round up to 7.

37 is not evenly divisible by 3, 4, 5, 6, and 7, so we can say confidently that it is prime. You can check it in the irb:

``````\$irb
> 37 % 2
=> 1
> 37 % 3
=> 1
> 37 % 4
=> 1
> 37 % 5
=> 2
> 37 % 6
=> 1
> 37 % 7
=> 2
``````

We did not get 0 as the remainder, so it is not evenly divisible by any of the numbers from 3,4,5,6, and 7. To double check our solution, we can check if a number is prime in Ruby. Go to irb :

``````> require 'prime'
=> true
> Prime.prime?(37)
=> true
``````

# Steps

## Step 1

Let's write our first test. We know we have to generate a list of divisors before we can check if the given number is prime or not by using the modulo operator. The first spec in findprimespec.rb will be something like:

``````describe FindPrime do
it 'should generate a list of divisors for the given number' do
# 1. Given a number 37
# 2. I expect the list of divisors to be [2,3,4,5,6,7]

end
end
``````

## Step 2

This is tough for a first test. Can we simplify this even further? How about we find the upper number? Change the first spec as follows:

``````describe FindPrime do
it 'should find the upper factor to test the prime' do
find_prime = FindPrime.new(37)

upper_factor = find_prime.upper_factor
expect(upper_factor).to eq(7)
end
end
``````

This is testing the implementation. We will come back to this issue later.

## Step 3

``````class FindPrime
def initialize(n)
@n = n
end

def upper_factor
Math.sqrt(@n)
end
end
``````

This fails with the error message 'expected 7 got 6.08'.

## Step 4

Change the upper_factor method as follows:

``````def upper_factor
Math.sqrt(@n).ceil
end
``````

The test now passes.

## Step 5

Now we are ready to tackle what we had in Step 1:

``````describe FindPrime do
it 'should generate a list of divisors for the given number' do
# 1. Given a number 37
# 2. I expect the list of divisors to be [2,3,4,5,6,7]

end
end
``````

The above spec now becomes the second spec as follows:

``````describe FindPrime do
it 'should generate a list of divisors for the given number' do
find_prime = FindPrime.new(37)

divisors_list = find_prime.divisors_list
expect(divisors_list).to eq([2,3,4,5,6,7])
end
end
``````

## Step 6

Add the divisors_list method to FindPrime class as follows:

``````class FindPrime
def initialize(n)
@n = n
end

def upper_factor
Math.sqrt(@n).ceil
end

def divisors_list
(2..upper_factor).to_a
end
end
``````

The test now passes.

## Step 7

Let's add the third spec as follows:

``````it 'should return true given a number input of 37' do
find_prime = FindPrime.new(37)

is_prime = find_prime.prime?
expect(is_prime).to eq(true)
end
``````

## Step 8

Based on what we learned in the solution domain analysis section, we can now add the prime? method to the FindPrime class as follows:

``````# 1. Get the list of factors to use for the modulo check
# 2. Find the modulo using the first number
# 3. Repeat modulo for all numbers
# 4. If all the numbers' modulo returns remainder, then return true otherwise false
def prime?
divisors_list.each do |e|
remainder = @n % e
return false if remainder == 0
end
return true
end
``````

## Step 9

Hide the divisors_list method by making it private. So the following test is not required because the method is now private and it gets indirectly tested.

``````it 'should generate a list of divisors for the given number' do
find_prime = FindPrime.new(37)

divisors_list = find_prime.divisors_list
expect(divisors_list).to eq([2,3,4,5,6,7])
end
``````

Delete that test. Now the entire listing looks like this:

``````class FindPrime
def initialize(n)
@n = n
end

def prime?
divisors_list.each do |e|
remainder = @n % e
return false if remainder == 0
end
return true
end

def upper_factor
Math.sqrt(@n).ceil
end

private

def divisors_list
(2..upper_factor).to_a
end
end

describe FindPrime do
it 'should find the upper factor to test the prime' do
find_prime = FindPrime.new(37)

upper_factor = find_prime.upper_factor
expect(upper_factor).to eq(7)
end

it 'should return true given a number input of 37' do
find_prime = FindPrime.new(37)

is_prime = find_prime.prime?
expect(is_prime).to eq(true)
end

end
``````

## Step 10

We can do the same to the upper_factor method. The listing now looks like this:

``````class FindPrime
def initialize(n)
@n = n
end

def prime?
divisors_list.each do |e|
remainder = @n % e
return false if remainder == 0
end
return true
end

private

def divisors_list
(2..upper_factor).to_a
end

def upper_factor
Math.sqrt(@n).ceil
end

end

describe FindPrime do

it 'should return true given a number input of 37' do
find_prime = FindPrime.new(37)

is_prime = find_prime.prime?
expect(is_prime).to eq(true)
end

end
``````

The first two tests we wrote is like scaffold of a building. Once the building construction is done, scaffold goes away.

## Step 11

Let's get rid of the magic number 2 by making it a constant. The FindPrime now looks like this:

``````class FindPrime
LOWER_BOUND = 2

def initialize(n)
@n = n
end

def prime?
divisors_list.each do |e|
remainder = @n % e
return false if remainder == 0
end
return true
end

private

def divisors_list
(LOWER_BOUND..upper_factor).to_a
end

def upper_factor
Math.sqrt(@n).ceil
end

end
``````

# Summary

In this article we saw how the tests that test the implementation eventually is deleted. We hide the implementation details by making it private to the class. Sometimes, you will have many private methods that operates on the same set of data and might be hiding an abstraction. In such cases, you can extract the functionality in the private methods into it's own class that can be tested separately.

# Exercises

1. Is it a good idea to check the data structure that is used for the implementation?
2. In the tests we have divisors_list that is checked with the elements in an Array. Is this a good idea? Why?

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