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Show that the square of any positive odd integer is of the form 4m+1 for some integer m.

Posted by Satheesh Kumar 7 years, 2 months ago

CBSE > Class 10 > Mathematics

1 answers

Gaurav Seth 7 years, 2 months ago

Let a be any positive integer and b = 4
Then, by Euclid''s algorithm a = 4q + r for some integer q 0 and 0 r < 4
Since, r = 0, 1, 2, 3
Therefore, a = 4q or 4q + 1 or 4q + 2 or 4q + 3

Since, a is an odd integer, o a = 4q + 1 or 4q + 3

Case I: When a = 4q + 1

Squaring both sides, we have,
a2 = (4q + 1)2
a2 = 16q2 + 1 + 8q
= 4(4q2 + 2q) + 1
= 4m + 1 where m = 4q2 + 2q

Case II: When a = 4q + 3

Squaring both sides, we have,
a2 = (4q +3)2
    = 16q2 + 9 + 24q
    = 16 q2 + 24q + 8 + 1
    = 4(4q2 + 6q + 2) +1
    = 4m +1 where m = 4q2 +7q + 2

Hence, a is of the form 4m + 1 for some integer m.

2Thank You

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