Question

Show that any positive even integer is of the form 4q, or 4q + 2, where q is some integer

Answer 1

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Answer 2

Step-by-step explanation:

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 that is (2×2q) is an even number.

Therefore, 4q+1 is an odd number.

Now, 4q+2 that is 2(2q+1) which is also an even number.

Therefore, (4q+2)+1=4q+3 is an odd number.

Hence, we can say that any even integer can be written in the form of 4q or 4q+2 where q is a whole number