Really Mr. Bond
Problem
115 2 = (110 x 120) + 25
that is 13225
895 2 = (890 x 900) + 25
that is 801025
Can you explain what is happening and generalise?
Getting Started
Is it always the case that when you square a number whose last digit is 5 you always end with 25?
By breaking the number down into a form (x + 5) it may then be possible to see what is happening and why.
Student Solutions
This is the solution sent in by Yatir Halevi. Thanks Yatir. A correct solution was also received from Andrei Lazanu.
Let's say we want to find the square of $a$
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.
For instance if we take $a=35$, we can take $b=5$, we get $35^2=(35+5)\times(35-5)+5^2 =40\times30+25 =1200+25 =1225$.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 "stolen" by the 25 that is added on.