SOLUTION: Prove : If m is an odd integer, then 4 divides m^2 +2m+5 I have gotten this so far but i dont know if its right and how to finish working it out! Proof: Let m be an odd integer

Question 527290: Prove : If m is an odd integer, then 4 divides m^2 +2m+5
I have gotten this so far but i dont know if its right and how to finish working it out!
Proof: Let m be an odd integer
there is an integer such that m=2k+1
then m^2 +2m +5 = (2k+1)^2 + 2(2k+1) +5
then m^2 + 2m +5= (4k^2 +4k +1) + (4k+2) +5
then m^2 +2m +5= (4k^2 +4k+4k) +1+2+5
then m^2 +2m +5 = 4(k^2 +k+k) + 8
Answer by richard1234(7193) (Show Source): You can put this solution on YOUR website!
You're correct. All you need to do now is factor it to 4(k^2 + 2k + 8). Since k^2 + 2k + 8 is an integer, the expression is divisible by 4.

Another way to prove it is to use modular arithmetic, or arithmetic dealing with remainders. If m is an odd integer, then m ≡ 1 mod 2 (i.e. m divided by 2 leaves a remainder 1. The ≡ sign means "equivalent"). Then m^2 ≡ 1 (mod) 4, 2m ≡ 2 (mod 4), and 5 ≡ 1 (mod 4). We can add modulos or residues the same way we add numbers; in this case, m^2 + 2m + 5 ≡ 1 + 2 + 1 ≡ 4 ≡ 0 (mod 4). Note that if a number is m mod n, we can subtract any multiple of n and still obtain an equivalent result. Since the expression is 0 mod 4, it is divisible by 4 (since it leaves remainder 0).
RELATED QUESTIONS
Conjecture: Find all positive integers 'a' such that there exists an integer 'm' with the (answered by richard1234)

The area of a rectangle is 24 square meters. Find the length and width of the rectangle... (answered by JBarnum)

Prove: if 4 divides mn, then m is even or n is even. i saw that this was p implies q... (answered by richard1234)

Problem is: {{{(8)/(2m+1)-(1)/(m-2)=(5)/(2m+1)}}} I've tried several ways, but can't... (answered by Alan3354,richwmiller)

If m is an odd integer, which expression always represents an odd integer? A) 3^m - 1 (answered by richard1234)

I have a problem where I am supposed to write each rational expression in lowest terms... (answered by Earlsdon)

There exists an integer 'a' such that if a|2m+1 and/or a|(m^2+1) and/or (answered by richard1234)

Good morning, I have been working on this problem..F(x)=sq rt of 2x-x/2 and x=t. I have... (answered by stanbon)

im doing "zero and negaive exponents"... in need to "use the quotient of powers property (answered by wuwei96815)