1783 /www/nrich/html/content/96/11/six4/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><p>Charlie is thinking of a number.<br></br> His number is both a multiple of $5$ and a multiple of $6$.<br></br> <strong>What could his number be?</strong></p> <p>Alison is thinking of a number.<br></br> Her number is a multiple of $4$, $5$ and $6$.<br></br> <strong>What could her number be?</strong></p> <p>Charlie is thinking of a number that is $1$ more than a multiple of $7$.<br></br> Alison is thinking of a number that is $1$ more than a multiple of $4$.<br></br> <strong>Could they be thinking of the same number?</strong></p> <p>Charlie is thinking of a number that is $3$ more than a multiple of $5$.<br></br> Alison is thinking of a number that is $8$ more than a multiple of $10$.<br></br> <strong>Could they be thinking of the same number?</strong></p> <p>Charlie is thinking of a number that is $3$ more than a multiple of $6$.<br></br> Alison is thinking of a number that is $2$ more than a multiple of $4$.<br></br> <strong>Could they be thinking of the same number?</strong></p> <p> </p> <p><a href="/content/96/11/six4/RemaindersCards.pdf">CARDS</a></p> <p> </p> <p>The interactivity below can be used to check your answers. It allows you to choose a divisor and then select numbers in one of the columns. <a href="/1783/clue">Here</a> are a couple of examples of how it can be used.<br></br> <br></br> <br></br> <mdo:flash height="400" id="/content/96/11/six4/Rems.swf" width="550"><param name="allowfullscreen" value="true"></param><param name="allowfullscreen" value="true"></param> <param name="movie" value="/content/96/11/six4/Rems.swf"></param> <param name="flashplayerversion" value="9"></param> </mdo:flash><br></br> <br></br> <a href="/content/96/11/six4/RemaindersCards.pdf">Here</a>  is a set of cards</p> <p> </p> <p> </p> <p><span style="font-weight: bold;">Now try out the problem generator below. When you click "Start" the computer will select at random an integer between 1 and 100. Can you identify the chosen number?</span><br></br> <br></br> You can use the interactivity above to help you, but eventually, try to identify the numbers without the aid of the interactivity.<br></br> <br></br> <br></br> <a href="/content/96/11/six4/Remainders.swf">Full Screen Version</a><br></br> <mdo:flash height="270" id="/content/96/11/six4/Remainders.swf" width="550"><param name="allowfullscreen" value="true"></param><param name="allowfullscreen" value="true"></param> <param name="movie" value="/content/96/11/six4/Remainders.swf"></param> <param name="flashplayerversion" value="9"></param> </mdo:flash></p> <p><span style="font-weight: bold;">One final question:</span></p> <p>We know that</p> <div style="margin-left: 40px;">When 59 is divided by 5, the remainder is 4<br></br> When 59 is divided by 4, the remainder is 3<br></br> When 59 is divided by 3, the remainder is 2<br></br> When 59 is divided by 2, the remainder is 1</div> <br></br> <p>Can you find the smallest number with the property that when it is divided by each of the numbers 2 to 10, the remainder is always one less than the number it is has been divided by? Don&#39;t forget to explain your reasoning.<br></br> <br></br>  </p></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p class="editorial">David from Mount Carmel School sent us his working to one of the problems:</p> The numbers which can be divided by five but leave a remainder of four are:<br></br> 4, 9, 14, 19, 24, 29, 34, 39, 44, 49, 54, 59, 64, 69, 74, 79, 84, 89, 94 and 99. <div>The numbers which can be divided by three but leave a remainder of two are:<br></br> 2, 5, 8, 11, 14...</div> <div>...wait,</div> <div>there it is!</div> <div>14!</div> <br></br> <div>To complete REMAINDERS, I took all the numbers which when divided by five left a remainder of four, and took all the numbers which when divided by three left a remainder of two.</div> <div>With these numbers, I then tried to match a number in one category with a number in the other category which was the same, and I got the answer!</div> <br></br> <div class="editorial">The final question asked you to find the smallest number with the property that when it is divided by each of the numbers 2 to 10, the remainder is always one less than the number it is has been divided by. Anurag from Queen Elizabeth's Grammar School in Horncastle explained how to solve it:</div> <br></br> <div>When 59 is divided by 5, the remainder is 4</div> <div>When 59 is divided by 4, the remainder is 3</div> <div>When 59 is divided by 3, the remainder is 2</div> <div>When 59 is divided by 2, the remainder is 1</div> <br></br> <div>The Lowest Common Multiple, or LCM, of 5, 4, 3 and 2 is 60.</div> <div>60 - 1 = 59.</div> <div>60 / 5 = 12.</div> <div>59 / 5 = 11 remainder 4.</div> <div>There is a remainder of 4 because we have subtracted the 1 that would be necessary to get a remainder of 0.</div> <br></br> <div>The same applies to the question we have been asked.</div> <div>We find the LCM of the numbers 2 to 10, and subtract 1.</div> <div>The LCM of 2, 3, 4, 5, 6, 7, 8, 9 and 10 is 2520</div> <div>(2 x 3 x 3 x 4 x 5 x 7 = 2520)</div> <br></br> <div>So the number we are looking for is 2520 - 1 = 2519</div> <br></br> <div><span class="editorial">Kien from Warminster School also sent us the solution to this question:</span></div> <br></br> <div>We have:</div> <div>x / 2 remainder 1</div> <div>x / 3 remainder 2</div> <div>x / 4 remainder 3</div> <div>....</div> <div>x/ 10 remainder 9</div> <br></br> <div>So, (x + 1) must divisible by 2, 3, 4, 5, 6, 7, 8, 9, 10.</div> <br></br> <div>So the smallest value for (x + 1) = 10 * 9 * 7 * 4 = 2520</div> <div>( the smallest number that has 2, 3, 4, 5, 6, 7, 8, 9 and 10 as factors)</div> <br></br> <div>So the smallest number that has the property is 2519.</div> <br></br> <div><span class="editorial">Kamal from Holy Angel School approached it like this:</span></div> <br></br> <div>Let N be the required number.</div> <br></br> <div>We can write,</div> <div>N = 2a + 1 = 2(a + 1) - 1</div> <div>N = 3b + 2 = 3(b + 1) - 1</div> <div>N = 4c + 3 = 4(c + 1) - 1</div> <div>N = 5d + 4 = 5(d + 1) - 1</div> <div>N = 6e + 5 = 6(e + 1) - 1</div> <div>N = 7f + 6 = 7(f + 1) - 1</div> <div>N = 8g + 7 = 8(g + 1) - 1</div> <div>N = 9h + 8 = 9(h + 1) - 1</div> <div>N =10i + 9 =10(i + 1) - 1</div> <br></br> <div>=&gt; (N + 1) is divisible by all of 2, 3, 4, 5, 6, 7, 8, 9 and 10.</div> <div>The smallest such number will be N = LCM of (2, 3, 4, 5, 6, 7, 8, 9, 10) - 1 = 2520 - 1 = 2519</div> <br></br> <span class="editorial">Well done to you all.</span><br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <h3>Why do this problem?</h3> This problem gives a context for the systematic search for solutions, looking for efficient strategies, and reflecting on which aspects of the problem causes it to be harder/easier. The task could provide an interesting context for practising routine tables facts. The random question generator could be used in lesson starters.<br></br> <br></br> <h3>Possible approach</h3> <div>You could start with a whole class counting activity:</div> <br></br> <div>Start counting together, speaking loudly on the numbers in the two times table, and quietly on the other numbers. Now split the class in two. Ask half the class to continue doing the same and ask the other half to only speak loudly on the numbers in the five times table.</div> <br></br> <div>Which numbers were <span style="font-weight: bold;">quiet</span> ?</div> <div>Which numbers were <span style="font-weight: bold;">fairly loud</span> and which were <span style="font-weight: bold;">very loud</span> ?</div> <br></br> <div>Now split the class in three. Two groups to continue as before and one group to only speak loudly on the numbers in the three times table.</div> <br></br> <div>Can they predict what they will hear?</div> <div>Which numbers will be quiet?</div> <div>Which numbers will be fairly loud and which will be very loud?</div> <br></br> <div>Try it.</div> <br></br> <div>Class could be split in four and the new group could be asked to speak loudly on the multiples of four.</div> <div>When will everyone speak loudly?</div> <br></br> <div>Start again and select two numbers which have a common factor, for example, 4s and 6s.</div> <div>Ask students to predict which numbers will be spoken loudly.</div> <div>Try it.</div> <br></br> <div>After this introductory activity, pose a question from the question generator. Give students a few minutes to think of numbers that fit one or more of the conditions. Gather some answers and explanations until the whole group feel confident suggesting numbers based on statements about divisors and remainders.</div> <br></br> With the same, or a new problem, ask students to work in pairs to find:<br></br> a number that fits all the conditions,<br></br> then to find all the numbers under 100 that fit them all,<br></br> then to write two sentences to explain how they know they have got them all.<br></br> <br></br> With the group together, ask for feedback, and put the answer into the answer box. If you have an interactive whiteboard, it might be appropriate to illustrate the logic with the coloured ball interactivity.<br></br> <br></br> Generate a selection of questions, ask students to pick out particular questions that seem easiest/hardest and work on those. On the board, write &quot;what makes a question like this easy/hard?&quot; and tell students that you'll be collecting suggestions after 15 mins.<br></br> <br></br> If a computer room is available, set students to work at computers in pairs. Students can use the coloured ball interactivity to help them, but emphasise that eventually you would like them to identify the numbers without the aid of the interactivity.<br></br> <br></br> You can print this <a href="/content/96/11/six4/Number%20grid.doc">10 by 10 number grid</a> so that students can keep a record of their working as they narrow down the possibilities.<br></br> <br></br> Then set the students to play <a href="http://nrich.maths.org/public/viewer.php?obj_id=6402&amp;part=">The Remainders Game</a><br></br> Who can reach 100 points in the least number of games?<br></br> Ask students to explain any strategies they have generated.<br></br> <br></br> Finally, ask them to have a go at the last question in the problem and emphasise that you will expect themto justify their conclusions.<br></br> <h3>Key questions</h3> <div>Which clues are most helpful?</div> <div>When does a clue provide no new information?</div> <div>What is the minimum number of divisions needed to identify the number?</div> <br></br> <h3>Possible extension</h3> <div>Students, working in pairs, could think of a number themselves and then give their partners three clues to help them identify their number, or, as in <a href="http://nrich.maths.org/public/viewer.php?obj_id=6402&amp;part=">The Remainders Game</a> they could allow their partner to choose the divisors.</div> <br></br> Students could also have a go at <a href="http://nrich.maths.org/public/viewer.php?obj_id=621&amp;part=">Ewa's Eggs</a><br></br> <br></br> <h3>Possible support</h3> <div><a href="http://nrich.maths.org/public/viewer.php?obj_id=5482&amp;part=">Clapping Times</a> and <a href="http://nrich.maths.org/public/viewer.php?obj_id=5483&amp;part=">Music to My Ears</a> suggest how the introductory activity could be extended.</div> <div>Use the coloured ball interactivity and ask students to predict what will happen.</div> <div><a href="http://nrich.maths.org/public/viewer.php?obj_id=1014&amp;part=">Flashing Lights</a> may be an accessible starting problem.</div> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p><strong>Example 1:</strong><br></br> Divide by 4 and select all the numbers in the right-hand column - they should all turn red.<br></br> Now divide by 5 and select all the numbers in the right-hand column - they should all turn yellow, but some will turn orange.<br></br> What is special about the numbers that turn orange?<br></br> Now divide by 3 and select all the numbers in the right-hand column - most should turn blue, but one will turn black.<br></br> What is special about the number that turns black?<br></br> What is special about the numbers that turn green and purple?</p> <p><strong>Example 2:</strong> (you will need to clear your previous work)<br></br> Find the numbers that have a remainder of 2 when divided by 5 - you&#39;ll need to divide by 5 and select the numbers in the second column.<br></br> Now select the numbers that have a remainder of 1 when divided by 2 (the odd numbers).<br></br> What is special about the numbers that turned orange this time?</p> <p>Try a few examples of your own and try to predict what will happen in each case.</p></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p>We seek a number <em>n</em> such that, when <em>n</em> is divided by each of the divisors 2,3,4,5...10, the remainder is one less than the divisor. This means that the number <em>n</em> +1 is exactly divisible by each of these numbers without a reminder and so <em>n</em> +1 is a multiple of all the numbers 2,3,4,5,...10. The smallest value of <em>n</em> +1 with the required property is 2x3x3x4x5x7=2520 so the smallest value of <em>n</em> is 2519.</p> <p>A correct solution was received from Heacham Middle School.</p> <br></br></mdoxml> 2 4 0 0 1 0 0 Remainders I'm thinking of a number. When my number is divided by 5 the remainder is 4. When my number is divided by 3 the remainder is 2. Can you find my number? Calculations and Numerical Methods Multiplication & division Admin Workshop Numbers and the Number System Factors and multiples Numbers and the Number System Divisibility