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The Modulo Operator For Finding Remainders From Division. by dkmathstats

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The Modulo Operator For Finding Remainders From Division.
<p style="text-align: left;">Hi there. This math post is on the modulo operator. The modulo operator finds the remainder from dividing two whole numbers together.&nbsp;</p><p><br></p><p style="text-align: center; "><img src="https://cdn.pixabay.com/photo/2015/09/13/10/06/pay-937884_1280.jpg" style="width: 527.5px;"><br></p><p style="text-align: center; "><a href="https://cdn.pixabay.com/photo/2015/09/13/10/06/pay-937884_1280.jpg" target="_blank" style="background-color: rgb(255, 255, 255); font-size: 1rem;">Pixabay Image Source</a><br></p><h2><u><span style="font-size: 24px;">Topics</span><span style="font-size: 24px;"></span></u></h2><p><br></p><ul><li>A Review Of Division</li><li>The Modulo Operator</li><li>Practice Problems</li><li>Answers To Practice Problems</li></ul><p><span style="font-size: 24px; font-weight: bold;"><br></span></p><p><span style="font-size: 24px; font-weight: bold;"><u>A Review Of Division</u></span><br></p><p><br></p><p><span style="font-size: 1rem;">Division is for figuring out how many groups can fit in a certain amount. When we say something like 24 divided by 4 is 6, we mean that from 24 objects we can have 6 groups with each group have 4 objects each. Note that with 24 divided by 4 we have a zero&nbsp; remainder as there are no objects left out.</span><br></p><p><span style="font-size: 1rem;"><br></span></p><p><span style="font-size: 1rem;"><b><u>Example One</u></b></span><br></p><p><span style="font-size: 1rem;">What is 100 divided by 10?</span><br></p><p><span style="font-size: 1rem;"><br></span></p><p><span style="font-size: 1rem;">From the number 100, how many 10s fit inside 100? The answer is 10. Alternatively, you can ask yourself from the multiplication viewpoint with what number multiplied by 10 gives 100?</span><br></p><p><br></p><p><b><u>Example Two</u></b></p><p><br></p><p>What is 28 divided by 3?</p><p><br></p><p>The number three fits into 28 at most 9 times. Nine multiplied by 3 is equal to 27. We cannot have ten groups of three as that would equal 30 which is more than 28.&nbsp;</p><p><span style="font-size: 1rem;">We can fit nine groups of three into 28. What is left over from nine groups of three is one. The remainder here is 1.</span><br></p><p><span style="font-size: 1rem;">The answer to 28 divided by 3 is 9 remainder 1. This is written as 9R1.</span><br></p><p><span style="font-size: 1rem;"><br></span></p><p><img src="https://cdn.pixabay.com/photo/2015/05/31/13/41/calculator-791831_1280.jpg" style="width: 527.5px;"><span style="font-size: 1rem;"><br></span></p><p style="text-align: center; "><a href="https://cdn.pixabay.com/photo/2015/05/31/13/41/calculator-791831_1280.jpg" target="_blank">Pixabay Image Source: Calculator</a></p><p style="text-align: center; "><span style="font-size: 1rem;"><br></span></p><h2><span style="font-size: 24px;"><u>The Modulo Operator</u></span></h2><p><span style="font-size: 1rem;">The modulo operator obtains the remainder from whole number division. It has applications in computer science/programming.</span><br></p><p><span style="font-size: 1rem;">In general if we have two whole numbers such as y and x where y is greater than or equal to x, we have this relation:</span><br></p><p><br></p><p style="text-align: center; "><i>y mod x = Remainder from y divided by x</i></p><p><br></p><p><b><u>Example One</u></b></p><p><span style="font-size: 1rem;">What is 100 mod 10?</span><br></p><p><span style="font-size: 1rem;"><br></span></p><p><span style="font-size: 1rem;">One hundred divided by ten gives ten with a zero remainder. Since there is a zero remainder, then the expression 100 mod 10 evaluates to 0.</span><br></p><p><br></p><p><b><u>Example Two</u></b></p><p><span style="font-size: 1rem;">What is 28 mod 3?</span><br></p><p><br></p><p>Recall that 28 divided by 3 (from example two in the previous section) is 9 with a remainder of one. The answer to 28 mod 3 is one.</p><p><br></p><p><span style="font-size: 1rem;"><b><u>Example Three</u></b></span><br></p><p><span style="font-size: 1rem;">Evaluate 33 mod 9.</span><br></p><p><br></p><p>From the number 33, you can create three groups of nine. Three groups of nine is equal to 27 (3 x 9 = 27). What is left over is 6. Evaluating 33 mod 9 gives 6.</p><p><br></p><p><img src="https://cdn.pixabay.com/photo/2016/06/25/12/52/laptop-1478822_1280.jpg" style="width: 527.5px;"></p><p style="text-align: center; "><a href="https://cdn.pixabay.com/photo/2016/06/25/12/52/laptop-1478822_1280.jpg" target="_blank">Pixabay Image Source</a><br></p><h2><span style="font-size: 24px;"><u>Practice Problems</u></span></h2><p><br></p><p><span style="font-size: 1rem;">Evaluate each of the following.</span><br></p><p><br></p><p><b>1)</b> 10 mod 10</p><p><br></p><p><b>2)</b> 11 mod 9</p><p><br></p><p><b>3)</b> 77 mod 25</p><p><br></p><p><b>4)</b> 8 mod 3</p><p><br></p><p><b>5) </b>55 mod 8</p><p><br></p><p><b>6) </b>Which has the greater number? Is it 83 mod 10 or 37 mod 8?</p><p><br></p><p><b>7) </b>Given that x is a whole positive number, what is x mod x? What is 2x mod x?</p><p><span style="font-size: 1rem;"><br></span></p><h2><span style="font-size: 24px;"><u>Answers To Practice Problems</u></span></h2><p><br></p><p><b>1)</b> 10 mod 10 = 0</p><p><br></p><p><b>2)</b> 11 mod 9 = 2</p><p><br></p><p><b>3) </b>77 mod 25 = 3</p><p><br></p><p><b>4)</b> 8 mod 3 = 2</p><p><br></p><p><b>5) </b>55 mod 8 = 7</p><p><br></p><p><b>6)</b> 83 mod 10 = 3 and 37 mod 8 = 5 so 37 mod 8 is greater than 83 mod 10.</p><p><br></p><p><b>7)</b> x mod x = 0 and 2x mod x = 0</p><p><br></p><p style="text-align: center; "><span style="font-size: 1rem;">Thank you for reading.</span></p><p style="text-align: center; "><span style="font-size: 1rem;"><br></span></p><p style="text-align: center; "><img src="https://cdn.pixabay.com/photo/2015/12/04/17/10/fractal-1076851_1280.jpg" style="width: 527.5px;"><span style="font-size: 1rem;"><br></span></p><p style="text-align: center; "><a href="https://cdn.pixabay.com/photo/2015/12/04/17/10/fractal-1076851_1280.jpg" target="_blank">Pixabay Image Source</a><span style="font-size: 1rem;"><br></span><br></p><div><br></div>
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