Hello guys

Big day today. Been thinking for some hours on this . Getting quite tired nearly to sleep . Again, really really kind of your help I will look other problems but for not so easy me .

PROBLEM

If $n\geq8$ show that any postage of $n$ cents can be made exactly usng only 3-cent & 5-cent stamps.

SOLUTION

The solution gives 2 rules and assert that for $k\geq8$"1 of these cases must occur "

One or more 5-cent stamps are used to make up $k$ cents postage.Rule 1.

Replace 1 of them with $2$ 3-cent stamps to make $k+1$ cents postage

Three or more 3-cent stamps are used to make up $k$ cents postage.Rule 2.

Replace 3 of them with $2$ 5-cent stamps to make $k+1$ cents postage.

~~~~~~

$k=8 \Rightarrow $ 3-cent, 5cent

$k=9 \Rightarrow $ only 3-cent

$k=10 \Rightarrow $ only 5-cent

$k=11 \Rightarrow $ k+1 withRule 1

$k=12 \Rightarrow $ only 3-cent

$k=13 \Rightarrow $ k+1 withRule 2

$k=14 \Rightarrow $ ????

QUESTION

so consider $k=14$

We have the case $14=5+9$ where we replace $2$ 5-cent with $3$ 3-cent. This is not either(replace $1$ 5 cent with $2$ 3-cent) orRule 1(replace $3$ 3-cent with $2$ 5-cent) . So maybe we needRule 2.Rule 3

Three or more 5-cent stamps are used to make $k+1$ cents postage.Rule 3.

Replace 2 of them with $3$ 3-cent stamps to make $k$ cents postage

Or maybeinRule 3orRule 1?Rule 2