4. Use Euclid's division lemma to show that the square of any positive integer is either of the form
3mor 3m - 1 for some integer m.
Answers
Hi Beautiful Soul,
Use Euclid division lemma to show that
the square of any positive integer is
either of the form 3m or 3m+1 for some
integer m.
Let a be any positive integer and b=3.
=) a = 3q +r, r=0 or 1 or 2.
(By Euclid's lemma)
=) a = 3q or 3q +1 or 3q + 2 for positive
integer q.
1st case,
If a = 3q:
=) a = (3q)2
= 9
= 3(3q)
= 3m, where m= 3q
2nd case,
If a = 3q+1,
=) a? = (39+1)
= (3q) + 2(39)(1) + 12
= = 9q + 6q +1
= 3(3q2 + 2q) + 1
= 3m + 1, where m = 3q2 + 2q.
3rd case,
If a=3q+2:
=) a? = (39+2)2
= (39)2 + 2(39)(2)+22
= 9q? + 12q + 4
= = 9q? + 12q + 3+1
= 3(3q2 +4q + 1) + 1
= 3m + 1, where m = 3q+49 +1.
Therefore, the square of any positive integer is either of the form 3m or 3m+1.
All the very best for your Bright Future ❤
Answer:
It is the correct answer.
Step-by-step explanation:
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