Question

If k is an integer then which of the following is true?

A) An integer of the form 4k+1 can always be put in the form 2k-1

B) An integer of the form 4k+3 can always be put in the form 2k+1

C) An integer of the form 2k-1 can always be put in the form 4k+1

D) An integer of the form 2k-1 can always be put in the form 4k+3

E) An integer of the form 2k+1 can always be put in the form 4k+3

Answer 1

Since an integer consists of all the values between (-∞,+∞)

Check by putting values ie negative , zero and a positive integer value,

A) 4k+1 will take values :

1 when k=0, [1 can be written as 2k+1 by putting k=0]

-7 when k=-2, [-7 can be written as 2k+1 by putting k=-4]

13 when k=3 [13 can be written as 2k+1 by putting k=6]

So this condition holds true.

Now take option E

E) An integer of the form 2k+1 can not always be put in the form 4k+3

eg- 13 can be written as 2k+1 by putting k =6, but it cannot be written as 4k+3

Similarly check for other values.