Using the division algorithm, show that

x^(3)+2=0

cannot have any integer solutions. (Note: you should have most of this problem completed from the in-class activity.)\ 1\ Let

a,b

, and

c

be integers such that

a|b

| and

a|c

|. Prove that, for any integers

u

and

v

,

a|ub+vc

|.\ Let

m

be any positive integer. Prove that if

r

is the reduction of

N

modulo

m

with

r!=0

, then

m-r

is the reduction of

-N

modulo

m

. (Note: you likely have a lot of this problem completed from the in-class activity!)

3. Using the division algorithm, show that x3+2=0 cannot have any integer solutions. (Note: you should have most of this problem completed from the in-class activity.) 1 4. Let a,b, and c be integers such that ab and ac. Prove that, for any integers u and v, aub+vc. 5. Let m be any positive integer. Prove that if r is the reduction of N modulo m with r=0, then mr is the reduction of N modulo m. (Note: you likely have a lot of this problem completed from the in-class activity!)