[OlymLSec_20150507BDP] A Partition of Unity via Factorisation

Question

Introduction

     A premier school in Singapore is said to have set a question similar to this question in a regular school test.  I suspect it was meant to select pupils to represent the school for mathematics Olympiad.
     I present two solutions: one using the idea of prime factor, the other one using the possible factorisation of some number into two factors.

Solution 1

Solution 2

Remark

     The Factorisation Grid helps us to perform the factorisation.  It is presented in the form that I learned it, in which the cross terms are written in the middle.  I prefer it this way.  Nowadays, the textbooks and the schools put these terms on a column on the right instead of in the middle.

     The word “unity” in mathematics is just a fancy word for the number  1.  It is a “partition” because the number is being split into two separate parts (fractions).  The “partition of unity” concept can be extended, and is useful in advanced mathematics including signal processing, weighted averages, spline functions and topology.

Suitable Levels

* Lower Secondary Mathematics Olympiad

* any student who loves a challenge in algebra or number theory