867 /www/nrich/html/content/02/09/six4/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><p>115 ² = (110 x 120) + 25<br></br> that is 13225</p> <p>895 ² = (890 x 900) + 25<br></br> that is 801025</p> <p>Can you explain what is happening and generalise?</p></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p>This is the solution sent in by Yatir Halevi. Thanks Yatir. A correct solution was also received from Andrei Lazanu.</p> <p>Let's say we want to find the square of $a$</p> <p>We know that $a^2 = a^2-b^2+b^2 = (a+b)\times(a-b)+b^2$and for every a, we can pick a certain b that will make the calculation$a^2$ as easy as possible.</p> For instance if we take $a=35$, we can take $b=5$, we get $35^2=(35+5)\times(35-5)+5^2 =40\times30+25 =1200+25 =1225$. <p>So, if$a$ is a number that ends with a 5: it can be written as $$a=10\times q + 5a^2=(10q+5)^2=(10q+5-5)\times(10q+5+5)+25=10q(10q+10)+25=10^2q(q+1)+25$$ So $a^2$is equal to $q(q+1)$ plus two zeros after it $(10^2)$ that are &quot;stolen&quot; by the 25 that is added on.</p> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> <p>Is it always the case that when you square a number whose last digit is 5 you always end with 25?</p> <p>By breaking the number down into a form (x + 5) it may then be possible to see what is happening and why.</p> </mdoxml> 2 5 0 0 0 1 0 115^2 = (11 x 12)x 25, that is 13225 895^2 = (89 x 90)x 25, that is 801025 Can you explain what is happening and generalise? Numbers and the Number System Number theory Numbers and the Number System Factors and multiples Numbers and the Number System Place value Algebra Index notation/Indices Numbers and the Number System Properties of numbers Algebra Manipulating algebraic expressions/formulae