Question

Show that an integer of the form 6k+5 is also of the form 3k+2, but not conversely

Answer 1

I think I can work on the first part, but have no idea on the second "converse part'

Answer 2

Can you show your work on the first part?

Answer 3

So if an integer is of the form 6k+5, then it is equivalent to 6k+3+2, which is equal to 3(2k+1)+2. Now if k is an integer 2k+1 is also an integer, so it is of the form 3k+2

Answer 4

I am sorry, I should have mentioned k is an integer.

Answer 5

Yes, then just try the reverse.
3k+2=6k+5-3k-3
=6k-3(k+1)+5
6{k-(k+1)/2}+5
=6(k-1)/2+5
(k-1)/2 may not be an integer if n is even.
See that the expression does not come out to be in the form 6n+5, where n is an integer. So the converse is true sometimes, but not always.

Answer 6

Ok?

Answer 7

for the converse...just give a specific counter example

Answer 8

2=3*0+2

2=6k+5
-3=6k
k=-1/2

which is not an integer