Question

1) IULITO
Use Euclid's division lemma, to show that the cube of ans positive integer 15
the form 3 or 3p+1 or 3p+2 for some integer 'p'

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Answer 1

≡QUESTION≡

Use Euclid’s division lemma to show that the square of any positive integer is either of the form 3p or 3p + 1 for some integer p.

                                                   

║⊕ANSWER⊕║

Let take a as any positive integer and b = 3.

Then using Euclid’s algorithm we get a = 3q + r  here r is remainder and value of q is more than or equal to 0  and r = 0, 1, 2 because 0 < r < b  and the value of b is 3 So our possible values will 3q+0 , 3q+1 and 3q+2

Now find the square of values  

Use the formula (a+b)² = a² + 2ab +b² to open the square bracket  

(3q)²  = 9q²  

if we divide by 3 we get no remainder  

(3q+1)²   = (3q)² + 2*3q*1  + 1²        

               =9q² + 6q +1

                = 3(3q² + 2q) +1

(3q+2)²  = (3q)² + 2*3q*2  + 2²  

               =9q² + 12q +4  

               = 3(3q² + 4q +1) +1 so we can write it in form of 3p +1 and value of m will 3q² + 4q +1

∴Square of any positive integer is either of the form 3p or 3p + 1 for some integer m.