Is a positive integer even if its square is even?

Let's take a positive integer n.We can write it as a product of prime numbers:n=p1p2...pr, where p1, p2, ..., pr are prime factors of n.Now, assume that n2 is even. Then one of the pi equals 2. Why?Note that n2=p12...pr2. Also, since n2 is even, then n2=2k for some k positive integer. => 2k=p12...pr2, which, since 2 is a prime, implies that 2 = pi for some i.Hence, n=2p1p2...pi-1pi+1...pr.And so, n is even. 

Answered by Andrei S. Maths tutor

2645 Views

See similar Maths University tutors

Related Maths University answers

All answers ▸

What does it mean for a matrix to be singular?


What is the difference between a Supremum and a Maximum of a sequence?


Differentiate f(x)= 3y^2 + ln (x) + sin x


Is there any rational number whose square is 2?


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 2013–2024

Terms & Conditions|Privacy Policy