using euclid division lemma to show that square of any positive integer is always equal to the 3m,3m+1
Answers
Answered by
2
Hey
let ' a' be any positive integer and b = 3.
we know, a = bq + r , 0 < r< b.
now, a = 3q + r , 0
the possibilities of remainder = 0,1 or 2
Case I - a = 3q
a2 = 9q2
= 3 x ( 3q2)
= 3m (where m = 3q2)
Case II - a = 3q +1
a2 = ( 3q +1 )2
= 9q2 + 6q +1
= 3 (3q2 +2q ) + 1
= 3m +1 (where m = 3q2 + 2q )
Case III - a = 3q + 2
a2 = (3q +2 )2
= 9q2 + 12q + 4
= 9q2 +12q + 3 + 1
= 3 (3q2 + 4q + 1 ) + 1
= 3m + 1 where m = 3q2 + 4q + 1)
From all the above cases it is clear that square of any positive integer ( as in this case a2 ) is either of the form 3m or 3m +1.
Hope this helps you ☺☺
let ' a' be any positive integer and b = 3.
we know, a = bq + r , 0 < r< b.
now, a = 3q + r , 0
the possibilities of remainder = 0,1 or 2
Case I - a = 3q
a2 = 9q2
= 3 x ( 3q2)
= 3m (where m = 3q2)
Case II - a = 3q +1
a2 = ( 3q +1 )2
= 9q2 + 6q +1
= 3 (3q2 +2q ) + 1
= 3m +1 (where m = 3q2 + 2q )
Case III - a = 3q + 2
a2 = (3q +2 )2
= 9q2 + 12q + 4
= 9q2 +12q + 3 + 1
= 3 (3q2 + 4q + 1 ) + 1
= 3m + 1 where m = 3q2 + 4q + 1)
From all the above cases it is clear that square of any positive integer ( as in this case a2 ) is either of the form 3m or 3m +1.
Hope this helps you ☺☺
Answered by
2
hello ,
Here is Ur answer .★
Hope this helps you.::::::::::::::::::::::::::::::::::::::::::::::
Sol .
Let A be any position integer and b= 3.
Than a = 3q+r for some Integer q ≥ 0
and r = 0, 1 , 2 because 0 ≤ r < 3
Therefore, a = 3q or 3q+1 or 3q +2 or,
where, k1 , k2 , or k3 are some positive integer.
Hence, it can be said that the square of any positive integer is either of the form 3m or 3m+1.
<<<☺>>>
Here is Ur answer .★
Hope this helps you.::::::::::::::::::::::::::::::::::::::::::::::
Sol .
Let A be any position integer and b= 3.
Than a = 3q+r for some Integer q ≥ 0
and r = 0, 1 , 2 because 0 ≤ r < 3
Therefore, a = 3q or 3q+1 or 3q +2 or,
where, k1 , k2 , or k3 are some positive integer.
Hence, it can be said that the square of any positive integer is either of the form 3m or 3m+1.
<<<☺>>>
Similar questions