Question

The set S consists of all odd multiples of 3: S = {..., -9, -3, 3, 9}. If the integers x and y are in S, then which of the following must also be in S?

a.) xy
b.) x +y
c.) x-y
d.) x/y
e.) -x - y

Answer 1

From advanced cal.

If S is odd multiples of 3 then...

S={ 3(2n+1)}, where n is an integer
also x and y belong to S

let y= 2n+1, x=2m+1

for a.) xy

3(2n+1)(3)(2m+1)
9(2n+1)(2m+1)
9(4nm+2n+2m+1)
9(2)[(nm+n+m)+1]
9(2k+1),
where k =nm+n+m, it also does not matter if k is even or odd since (2k+1) will always be odd

so... 9(2k+1)
3(3)(2k+1)
3(6k+3)

end of proof... 6k+3 is always odd and it is a multiple of 3, it meets the requirments

now for b.)

x+y
3(2n+1) + 3(2m+1)
3[(2n+1)+(2m+1)]
3[2m+2n+2]
3[(2)(m+n+1) where m+n=k
3[2k+1] 2k+1 is always odd and its being multiplied by 3

end of proof.

use the same idea with the c, d, ect.