<h2>Identifying Prime and Composite Numbers</h2>
<p>Some numbers, like \(72,\) have many factors. Other numbers, such as \(7,\) have only two factors: \(1\) and the number. A number with only two factors is called a <strong>prime number</strong>. A number with more than two factors is called a <strong>composite number</strong>. The number \(1\) is neither prime nor composite. It has only one factor, itself.</p>
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<h3>Prime Numbers and Composite Numbers</h3>
<p>A prime number is a counting number greater than \(1\) whose only factors are \(1\) and itself.</p>
<p>A composite number is a counting number that is not prime.</p>
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<p>The figure below lists the counting numbers from \(2\) through \(20\) along with their factors. The highlighted numbers are prime, since each has only two factors.</p>
<div class="wp-caption alignnone" style="width: 741px"><img alt="This figure shows a table with twenty rows and three columns. The first row is a header row. It labels the columns as “Number”, “Factor” and “Prime or composite?” The second row lists the number 2, in red, under the “Number” column, the numbers 1 and 2 under the “Factors” column and the word prime under the “Prime or Composite?” column. The third row lists the number 3, in red, under the “Number” column, the numbers 1 and 3 under the “Factors” column and the word prime under the “Prime or Composite?” column. The fourth row lists the number 4 under the “Number” column, the numbers 1, 2 and 4 under the “Factors” column and the word composite under the “Prime or Composite?” column. The fifth row lists the number 5, in red, under the “Number” column, the numbers 1 and 5 under the “Factors” column and the word prime under the “Prime or Composite?” column. The sixth row lists the number 6 under the “Number” column, the numbers 1, 2, 3 and 6 under the “Factors” column and the word composite under the “Prime or Composite?” column. The seventh row lists the number 7, in red, under the “Number” column, the numbers 1 and 7 under the “Factors” column and the word prime under the “Prime or Composite?” column. The eighth row lists the number 8 under the “Number” column, the numbers 1, 2, 4 and 8 under the “Factors” column and the word composite under the “Prime or Composite?” column. The ninth row lists the number 9 under the “Number” column, the numbers 1, 3 and 9 under the “Factors” column and the word composite under the “Prime or Composite?” column. The tenth row lists the number 10 under the “Number” column, the numbers 1, 2, 5 and 10 under the “Factors” column and the word composite under the “Prime or Composite?” column. The eleventh row lists the number 11, in red, under the “Number” column, the numbers 1 and 11 under the “Factors” column and the word prime under the “Prime or Composite?” column. The twelfth row lists the number 12 under the “Number” column, the numbers 1, 2, 3, 4, 6 and 12 under the “Factors” column and the word composite under the “Prime or Composite?” column. The thirteenth row lists the number 13, in red, under the “Number” column, the numbers 1 and 13 under the “Factors” column and the word prime under the “Prime or Composite?” column. The fourteenth row lists the number 14 under the “Number” column, the numbers 1, 2, 7 and 14 under the “Factors” column and the word composite under the “Prime or Composite?” column. The fifteenth row lists the number 15 under the “Number” column, the numbers 1, 2, 3, 5 and 15 under the “Factors” column and the word composite under the “Prime or Composite?” column. The sixteenth row lists the number 16 under the “Number” column, the numbers 1, 2, 4, 8 and 16 under the “Factors” column and the word composite under the “Prime or Composite?” column. The seventeenth row lists the number 17, in red, under the “Number” column, the numbers 1 and 17 under the “Factors” column and the word prime under the “Prime or Composite?” column. The eighteenth row lists the number 18 under the “Number” column, the numbers 1, 2, 3, 6, 9 and 18 under the “Factors” column and the word composite under the “Prime or Composite?” column. The nineteenth row lists the number 19, in red, under the “Number” column, the numbers 1 and 19 under the “Factors” column and the word prime under the “Prime or Composite?” column. The twentieth row lists the number 20 under the “Number” column, the numbers 1, 2, 4, 5, 10 and 20 under the “Factors” column and the word composite under the “Prime or Composite?” column." height="226" src="https://data.ulearngo.com/assets/4609d5d4-048e-4232-8ec1-87567c97547b" width="741"/><p class="wp-caption-text">Factors of the counting numbers from \(2\) through \(20,\) with prime numbers highlighted</p></div>
<p>The prime numbers less than \(20\) are \(2,3,5,7,11,13,17,\text{and}\phantom{\rule{0.2em}{0ex}}19.\) There are many larger prime numbers too. In order to determine whether a number is prime or composite, we need to see if the number has any factors other than \(1\) and itself. To do this, we can test each of the smaller prime numbers in order to see if it is a factor of the number. If none of the prime numbers are factors, then that number is also prime.</p>
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<h3>Determine if a number is prime.</h3>
<ol type="1">
<li>Test each of the primes, in order, to see if it is a factor of the number.</li>
<li>Start with \(2\) and stop when the quotient is smaller than the divisor or when a prime factor is found.</li>
<li>If the number has a , then it is a . If it has no prime factors, then the number is prime.</li>
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<h2>Example</h2>
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<p>Identify each number as prime or composite:</p>
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<li>\(\phantom{\rule{0.2em}{0ex}}83\phantom{\rule{0.2em}{0ex}}\)</li>
<li>\(\phantom{\rule{0.2em}{0ex}}77\)</li>
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<h3 data-type="title">Solution</h3>
<p>Test each prime, in order, to see if it is a factor of \(83\), starting with \(2,\) as shown. We will stop when the quotient is smaller than the divisor.</p>
<table data-label="" summary="The figure shows a table with six rows and three columns. The first row is a header row and labels the rows “Prime”, “Test”, and “Factor of 83?” Under the “Prime” column, the rows show the numbers 2, 3, 5, 7, and 11. Under the “Test” column, the second row lists “Last digit of 83 is not 0, 2, 4, 6, or 8”. Under the “Factor of 83?” column, all of the rows list “No.” Under the “Test” column, the third row lists “eight plus three equals eleven, and eleven is not divisible by three.”  Under the “Test” column, the fourth row lists “The last digit of eighty-three is not five or zero.” Under the “Test” column, the fifth row lists “eighty-three divided by seven equals eleven point eight, five, seven, ellipses.” Under the “Test” column, the sixth row lists “eighty-three divided by eleven equals seven point five four five ellipses.”">
<thead>
<tr>
<th><strong>Prime</strong></th>
<th><strong>Test</strong></th>
<th><strong>Factor of</strong> \(83?\)</th>
</tr>
</thead>
<tbody>
<tr>
<td>\(2\)</td>
<td>Last digit of \(83\) is not \(0,2,4,6,\text{or}\phantom{\rule{0.2em}{0ex}}8.\)</td>
<td>No.</td>
</tr>
<tr>
<td>\(3\)</td>
<td>\(8+3=11,\) and \(11\) is not divisible by \(3.\)</td>
<td>No.</td>
</tr>
<tr>
<td>\(5\)</td>
<td>The last digit of \(83\) is not \(5\) or \(0.\)</td>
<td>No.</td>
</tr>
<tr>
<td>\(7\)</td>
<td>\(83÷7=11.857\text{….}\)</td>
<td>No.</td>
</tr>
<tr>
<td>\(11\)</td>
<td>\(83÷11=7.545\text{…}\)</td>
<td>No.</td>
</tr>
</tbody>
</table>
<p>We can stop when we get to \(11\) because the quotient \(\text{(7.545…)}\) is less than the divisor.</p>
<p>We did not find any prime numbers that are factors of \(83,\) so we know \(83\) is prime.</p>
<p>Test each prime, in order, to see if it is a factor of \(77.\)</p>
<table data-label="" summary="The figure shows a table with five rows and three columns. The first row is a header row and labels the rows “Prime”, “Test”, and “Factor of 77?” Under the “Prime” column, the rows show the numbers 2, 3, 5, and 7. Under the “Test” column, the second row lists “Last digit is not 0, 2, 4, 6, or 8”. Under the “Factor of 77?” column, the first three rows list “No” and the fourth row lists “Yes.” Under the “Test” column, the third row lists “seven plus seven equals fourteen, and fourteen is not divisible by three.”  Under the “Test” column, the fourth row lists “The last digit is not five or zero.” Under the “Test” column, the fifth row lists “seventy-seven divided by eleven equals seven.”">
<thead>
<tr>
<th><strong>Prime</strong></th>
<th><strong>Test</strong></th>
<th><strong>Factor of \(77?\)</strong></th>
</tr>
</thead>
<tbody>
<tr>
<td>\(2\)</td>
<td>Last digit is not \(0,2,4,6,\text{or}\phantom{\rule{0.2em}{0ex}}8.\)</td>
<td>No.</td>
</tr>
<tr>
<td>\(3\)</td>
<td>\(7+7=14,\) and \(14\) is not divisible by \(3.\)</td>
<td>No.</td>
</tr>
<tr>
<td>\(5\)</td>
<td>the last digit is not \(5\) or \(0.\)</td>
<td>No.</td>
</tr>
<tr>
<td>\(7\)</td>
<td>\(77÷11=7\)</td>
<td>Yes.</td>
</tr>
</tbody>
</table>
<p>Since \(77\) is divisible by \(7,\) we know it is not a prime number. It is composite.</p>
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Identifying Prime and Composite Numbers

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Identifying Prime and Composite Numbers

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