Prove the square of an odd number is always 1 more than a multiple of 4

We will start by representing an odd number using algebra. Let n be a integer, i.e. 1,2,3,4,etc. then we can define an odd number as (2n)-1. This is because if a number is of the form 2n it must be even, since it can be divided by 2 and still be a whole number. So, if we take 1 away from the even number, i.e. 2n-1, then it must be odd.
Now we will square our odd number. (2n-1)2 = 4n2-4n+1 =4(n2-n)+1.The first term here 4(n2-n) is clearly a multiple of 4 since we have a 4 outside the brackets. We still have the 1 left over, so we have that the square of an odd number is always 1 more than a multiple of 4.

TD
Answered by Tutor285427 D. Maths tutor

31697 Views

See similar Maths GCSE tutors

Related Maths GCSE answers

All answers ▸

how to convert a decimal into a fraction, let the decimal be 0.75


Show the curve y = 4x^2 + 5x + 3 and the line y = x + 2 have exactly one point of intersection


Solve the equation "x^2 + 3x - 4 = 0".


y = x^2 + 4x + 7 Find the turning point of the equation by completing the square.


We're here to help

contact us iconContact usWhatsapp logoMessage us on Whatsapptelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

© MyTutorWeb Ltd 2013–2025

Terms & Conditions|Privacy Policy
Cookie Preferences