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Show that any positive odd integer is of the form 4q+1 or 4q+3 by using Euclid's division lemma where q is some integer

Posted by Shreedhar J 5 years, 1 month ago

CBSE > Class 10 > Mathematics

2 answers

Yogita Ingle 5 years, 1 month ago

Let a be any positive integer and b = 4. Then by Euclid’s algorithm,
a = 4q + r for some integer q ≥ 0, and r = 0, 1, 2, 3
So, a = 4q or 4q + 1 or 4q + 2 or 4q + 3 because 0 ≤ r < 4
Now, 4q i.e., 2(2q) is an even number
∴4q + 1 is an odd number.
4q + 2 i.e., 2(2q + 1) which is also an even number.
∴ (4q + 2) + 1 = 4q + 3 is an odd number.
Thus, we can say that any odd integer can be written in the form 4q + 1 or 4q + 3 where q is some integer.

1Thank You

Shreedhar J 5 years, 1 month ago

By Euclid division lemma, a=4q+r r=0,1,2,3 Case(i).r=0, a=4q=2(2q)=2m[even] (ii). r=1,a=4q+1=2(2q)+1=2m+1[even]

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