State precisely but concisely what it means for an integer n

State precisely (but concisely) what it means for an integer n to be divisible by 4. b. Is 0 divisible by 4? Justify your answer. c. Prove that if n is any odd integer, then n^2 has the form 4p + 1 for some integer P. d. Prove that if n is any odd integer, then n^2 has the form 8m+ 1 for some integer m.

Solution

a. An integer n is divisible by 4 if : n=4k for some integer k

b. Yes

0=0*4

c.

n is odd integer so, n=2k+1 for some integer k

n^2=(2k+1)^2=4k^2+4k+1=4(k^2+k)+1

Hence, n^2=4p+1, p=k^2+k

d.

n is odd so two cases

Case 1: n=4k+1 for some integer k

n^2=(4k+1)^2=16k^2+8k+1=8(2k^2+k)+1=8m+1 , m=2k^2+k

Case 2: n=4k+3 for some integer k

n^2=(4k+3)^2=16k^2+24k+9=8(2k^2+3k+1)+1

Hence proved