544 /www/nrich/html/content/97/12/six5/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p>Find the number which has 8 factors, such that the product of all the factors is 331776.</p> <p><em>Remember: A a factor (or divisor) is a number that divides exactly into another number.</em></p> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p>The solution is based on the divisors of 24.</p> <p>i.e 1 x 2 x 3 x 4 x 6 x 8 x 12 x 24 = 331776</p> <p><strong><em class="editorial" style="font-style: normal;">Alison at Maidstone Girls Grammar School arrived at her solution</em></strong> <span class="editorial">by trial and improvement techniques and had found that the divisors of 30 were too big when multiplied together and the divisors of 20 too small. She had ascertained that 24 did have quite a lot of divisors compared to others in this range.</span></p> <br></br> <div><strong><em class="editorial" style="font-style: normal; font-weight: 400;"><span style="font-weight: bold;">Serendipity helped Ashley and John from the Simon Langton Grammar School who &quot;luckily chose the right number (24)&quot;</span> .</em></strong> <span class="editorial">But, when using prime factors, they found that 24 = 2 x 2 x 2 x 3 and they noticed that numbers whose prime factors are of the form a x a x a x b have ONLY 8 divisors as they showed in a multiplication table, reproduced below:</span></div> <table> <tbody> <tr> <td style="font-style: italic; text-align: center;">x</td> <td style="font-weight: bold; font-style: normal; text-align: center;">1</td> <td style="font-weight: bold; font-style: normal; text-align: center;">a</td> <td style="font-weight: bold; font-style: normal; text-align: center;"> aa</td> <td style="font-weight: bold; font-style: normal; text-align: center;"> aaa</td> </tr> <tr> <td style="font-weight: bold; font-style: normal; text-align: center;">1</td> <td style="font-style: normal; text-align: center;">1</td> <td style="font-style: normal; text-align: center;">a</td> <td style="font-style: normal; text-align: center;">aa</td> <td style="font-style: normal; text-align: center;">aaa</td> </tr> <tr> <td style="font-weight: bold; font-style: normal; text-align: center;">b</td> <td style="font-style: normal; text-align: center;">b</td> <td style="font-style: normal; text-align: center;">ab</td> <td style="font-style: normal; text-align: center;">aab</td> <td style="font-style: normal; text-align: center;">aaab</td> </tr> </tbody> </table> <div><span class="editorial">To find the answer is then just a matter of substituting different prime numbers for a and b</span> .</div> <br></br> <div class="editorial">Similarly Robert (Smithdon High School) had relied upon the prime factorisation of 331776 for his solution.</div> <p><strong><em class="editorial" style="font-style: normal;">Gareth rom Hethersett High School, Norwich used algebra and found the fourth root of 331776, namely 24 - sort of hidden inside:</em></strong></p> <p>Let a, b, c, d, e, f, g, and h be the divisors where a x b x c x d x e x f x g x h = 331776 and a = 1.</p> <p>But a x h = h; b x g = h; c x f = h and d x e = h</p> <p>hence h x h x h x h = 331776</p> <p>and so h = 24</p> <hr></hr> <br></br> <br></br></mdoxml> 2 4 0 0 1 0 0 Find the number which has 8 divisors, such that the product of the divisors is 331776. Numbers and the Number System Factors and multiples Calculations and Numerical Methods Multiplication & division Numbers and the Number System Properties of numbers Numbers and the Number System Divisibility