Use the definitions of even odd prime and composite to justi

Use the definitions of even, odd, prime, and composite to justify the following. Assume r and s are particular integers. Is 4rs even? Is 6r + 4s^2 + 3 odd? If r and s are both positive. Is r^2 + 2rs + s^2 composite? Prove that for all integers m if m is even then 3 m + 5 is odd. Show that 0.5656565656.....is a rational number/

Solution

1.

a. Yes 4rs is a multiple of 4 and hence multiple of 2

b.

6r+4s^2+3=6r+4s^2+2+1=2(3r+2s^2+1)+1

Hence hte number is of the form: 2K+1 where ,K is integer

Hence number is odd

c.

Yes

r^2+2rs+s^2=(r+s)^2

So, r+s is a factor

2.

3m+5=m+2m+4+1=m+1+2m+4

If m is even then m+1 is odd, 2m+4 is even integer for all m

And sum of odd and even integers is odd

So,m+1+2m+4 is odd

3.

Let,

x=0.565656.......

Multiplying by 100 gives

100x=0.565656.....

Subtracting gives

99x=56

x=56/99

Hence proved