Prime Numbers

If a number can be divided into equal groups, then it is not a Prime number. It is a Composite number.

If a number cannot be divided into equal groups, then it is a Prime number.

Keep in mind: 

• groups cannot be of size one.
• groups cannot have fractional quantities like \(\frac{1}{2}\) in each group.

Do you think that you can divide things into groups with an equal number of things?

For example, we can divide \(12\) circles into groups with an equal number of circles in different ways:

This is because \(12\) can be factorized in the following ways:

• \(12 = 4 \times 3\)
• \(12 = 6 \times 2\)

We know that \(12\) can be factorized in the following ways as well:

• \(3\) groups of \(4\) because \(3 \times 4 =12\)
• \(2\) groups of \(6\) because \(2 \times 6 =12\)

(1 group of 12 or 12 groups of 1 don't count in our current system)

Quick recall: \(1, 2, 3, 4, 6\) and \(12\) are called factors of \(12\). 

So what we understood from the above example is that we can divide a number into groups with equal numbers of items/elements only if it can be factorized as the product of two numbers. 

But it is not possible to divide all numbers into groups with equal numbers. 

For example, \(7\) cannot be divided into groups of equal numbers. 

This is because \(7\) can only be factorized as follows:

• \(7 \times 1 = 7\)
• \(1 \times 7 = 7\)

Quick recall: \(1\) and \(7\) are the only factors of \(7\).

In the above examples:

• \(12\) is NOT a prime number (it means it is a composite number) because it could be divided into groups of equal numbers).
• \(7\) is a prime number because it could NOT be divided into groups of equal numbers.

Any whole number greater than \(1\) that has exactly two factors, \(1\) and itself, is defined as a prime number.

Another equivalent definition is:

Any whole number greater than \(1\) that is divisible only by \(1\) and itself, is defined as a prime number.

Any number greater than 1 that is not a prime number, is defined to be a composite number. Thus composite numbers will always have more than 2 factors.

Any whole number greater than 1 is either a prime or composite number.

For example:

• The number \(15\) has more than two factors: \(1, 3, 5,\) and \(15\). 
  Hence it is a composite number. 
• On the other hand, \(13\) has just two factors: \(1\) and \(13\).
  Hence it is a prime number.

Is 1 a Prime Number?

• 1 is neither prime nor composite, as the definition of prime numbers talks only about the whole numbers that are greater than \(1\).
• \(1\) has only one factor. So it cannot be a prime number as a prime number should have exactly two factors.
• \(1\) has only one factor. So it cannot be a composite number as a composite number should have more than two factors.

Characteristics of Prime Numbers

The characteristics of prime numbers are:

• A prime number is a whole number greater than \(1\).
• A prime number has exactly two factors, i.e. \(1\) and itself.

Prime and Composite Numbers

A prime number is a whole number greater than \(1\) that has EXACTLY two factors.

Example:

\(5\) can be factorized in only one way. i.e., \(1 \times 5\) (OR) \(5 \times 1\).

It has only two factors \(5\) and \(1\). 

So \(5\) is a prime number.

A composite number is a whole number greater than \(1\) that has MORE THAN two factors.

Example:

\(4\) can be factorized in multiple ways.

So the factors of \(4\) are \(1, 2 \text{ and } 4\). It has more than two factors. 

So \(4\) is a composite number.

There are \(25\) prime numbers from \(1\) to \(100\). 

The prime numbers between \(1\) to \(100\) are:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

These numbers are listed in the following table:

Prime numbers between 1 to 10	\( 2, 3, 5, 7\)
Prime numbers between 11 to 20	\( 11, 13, 17, 19\)
Prime numbers between 21 to 30	\( 23, 29\)
Prime numbers between 31 to 40	\(31, 37\)
Prime numbers between 41 to 50	\(41, 43, 47\)
Prime numbers between 51 to 100	\(53, 59, 61, 67, 71,\) \(73, 79, 83, 89, 97\)

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Here is the prime numbers chart from \(1\) to \(100\).

Here is how we can find the list of prime numbers between \(1\) to \(100\). 

We just strike off 1 as it is NOT a prime number. Then we strike off all the multiples of other numbers.

Then, the remaining numbers are the prime numbers.

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We cannot determine the largest prime number. 

The largest known prime number is \(2^{82,589,933}\)-1 having \(24,862,048\) digits.

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A pair of prime numbers that differ by \(2\) are called twin prime numbers. 

i.e. two prime numbers with a composite number in between are called twin primes.

For example:

1.  \(3\) and  \(5\) are twin primes. 
2.  \(5\) and  \(7\) are twin primes. 
3.  \(11\) and  \(13\) is another pair of twin primes.

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If a pair of numbers has no common factor apart from \(1\), then they are called coprime numbers. 

Example 1:

\(5\) and \(9\) are co-primes. That’s because:

Factors of \(5\) are \(1\) and \(5\). 

Factors of \(9\) are \(1, 3\) and \(9\). 

Here \(5\) and \(9\) have no common factors apart from \(1\), making them coprime.

Example 2:

\(6\) and \(10\) are NOT co-primes. That’s because:

Factors of \(6\) are \(1,2,3\) and \(6\). 

Factors of \(10\) are \(1, 2,5\) and \(10\). 

Here \(6\) and \(10\) have a common factor \(2\), which is apart from \(1\), making them NOT coprime.
 

If you would like to know whether any given number is prime, follow these steps:

• Write all the factors of the given number.
• Count the factors.
• If the number of factors is EXACTLY \(2\) then the given number is prime. 

If the number of factors is more than \(2\) then the given number is NOT prime (in that case, it is called a composite number).

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Example 1

 

 

Is \(6\) prime or NOT?

Solution:

Given number is \(6\).

The factors of \(6\) are  \(1, 2, 3, \text{ and }6\).

Thus \(6\) has \(4\) factors.

Since the number of factors of \(6\) is NOT exactly 2, it is NOT a prime number (so \(6\) is a composite number).

∴ \(6\) \(is\) \(NOT\) \(a\) \(prime\) \(number\).

Example 2

 

 

Is \(37\) prime or NOT?

Solution:

Given number is \(37\).

The factors of \(37\) are \(1 \text{ and } 37\).

Thus \(37\) has exactly two factors.

Since the number of factors of \(37\) is exactly 2, it is a prime number.

∴ \(37\) \(is\) \(a\) \(prime\) \(number\).

Example 3

 

 

Is \(20\) prime or NOT?

Solution:

Given number is \(20\).

The factors of \(20\) are \(1, 2, 4, 5, 10, \text{ and } 20\).

Thus \(20\) has more than two factors.

Since the number of factors of \(20\) is NOT exactly \(2\), it is NOT a prime number (So \(20\) is a composite number).

∴ \(20\) \(is\) \(NOT\) \(a\) \(prime\) \(number\).