Show that any positive odd integer is of the form 4q + 1 or 4q + 3, where q is some integer

Solution:

We will use Euclid’s division algorithm to show that positive odd integers are of the form 4q + 1, 4q + 3

Let ‘a’ be a positive odd integer.

a = bq + r where the value of r is 0 ≤ r < b.

⇒ b = 4 and where q is the quotient.

Given 0 ≤ r < 4, the remainders could be 0, 1, 2 and 3.

a = 4q + 0/ 4q or 

a = 4q + 1 or 

a = 4q + 2 or 

a = 4q + 3.

As we know that 4q is divisible by 2, it is an even integer.

Similarly, 4q + 2 is also divisible by 2, it is also an even integer.

Thus, 4q + 1, 4q + 3 are only odd integers if 0 ≤ r < 4

☛ Check: NCERT Solutions for Class 10 Maths Chapter 1

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Show that any positive odd integer is of the form 4q + 1 or 4q + 3, where q is some integer

Summary:

Therefore, any positive odd integer is of the form 4q + 1 or 4q + 3, where q is some integer and 0 ≤ r < 4

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