Or search by topic
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^2and for every a, we can pick a certain b that will make the calculationa^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, ifa 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
Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.
a) A four digit number (in base 10) aabb is a perfect square. Discuss ways of systematically finding this number. (b) Prove that 11^{10}-1 is divisible by 100.