Question

Please show all the steps and explain.

Prove that every amount of postage of 18 cents or more can be formed using just 4-cent and 7-cent stamps

Answer 1

P(n): Every stamp of amount cents can be made using 4 and 7 cent stamps

18

Base case: 18 can be made as

2 7 4 18

Induction assumption: We can make stamps of using just 4 and 7 cent stamps

18 <nくん

We will show that we can also made a stamp of cents

Note that can be made using 4 and 7 cent stamps means there exist non-negative a and b such that
k4a 7b

So that is a valid way of making k+1 cent stamp

k14a 7b1 4a7b2.4 7 4(a 2)7(b- 1)

Unless $b=0$ in which case
k4a
in which we can write
k14a 1 4a 3.7-5.4 4(a -5)+3 7

Which is a valid way of making k+1 stamps unless $a\le 4$

Note that both $b=0$ and $a\le 4$ are not possible because then
k4a 76 <16
contradicting $k\geq 18$

So that we can always make a stamp of k+1 cents

So that by induction, we can always make a stamp of n cents when using 4-cent and 7-cent stamps

18

$\blacksquare$

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