Math Postage Stamp Problem

Larry has 4-cent stamps and 9-cent stamps, which he can combine to produce various amounts of postage. For example, he can make 40 cents by using four 9-cent stamps and a 4-cent stamp, or by using ten 4-cent stamps. However, there are some amounts of postage he can't make exactly, such as 10 cents. {nl} {nl} What is the largest number of cents that Larry cannot make exactly from a combination of 4- and/or 9-cent stamps? {nl} {nl} Explain how you know your answer is correct. (You should explain two things: why Larry can't make the amount of your answer, and why he can make any bigger amount.)

All figures can be divided into 4 groups.

Group 1: 4n

Group 2: 4n+1

Group 3: 4n+2

Group 4: 4n+3

With 4-cent stamps we get Group 1. (Beginning with n=1 we have 4, 8, 12, 16, 32, ...)

With one 9-cent(4*2+1) we can start Group 2. (Beginning with n=2 we have 9 ) and continuing 4*3 + 1(=13) with one 4-cent, 4*4 + 1(=17) with two 4-cent, 4*5+1(=21) with three 4-cent etc.

With two 9-cent(4*4+2) we can start Group 3. (Beginning with n=4 we have 18 ) and continuing 4*5 + 2(=22) with one 4-cent, 4*6 + 2(=26) with two 4-cent, 4*7+2 (=30) with three 4-cent etc.

With three 9-cent(4*6+3) we can start Group 4. (Beginning with n=6 we have 27 ) and continuing 4*7 + 3(=31) with one 4-cent, 4*8 + 3(=35) with two 4-cent, 4*9+3 (=39) with three 4-cent etc.

$\begin{array}{|r|r|r|r|r|} \hline \text{All Numbers} \\ \hline \text{Group 1} &\text{Group 2} &\text{Group 3} &\text{Group 4} \\ 4n+1 & 4n+2 & 4n+3 & 4n \\ \hline 1 & 2 & 3 & 4\checkmark \\ & & & & +4 \\ 5 & 6 & 7 & 8\checkmark \\ & & & & +4 \\ 9\checkmark & 10 & 11 & 12\checkmark \\ & & & & +4 \\ 13\checkmark & 14 & 15 & 16\checkmark \\ & & & & +4 \\ 17\checkmark & 18\checkmark & 19 & 20\checkmark \\ & & & & +4 \\ 21\checkmark & 22\checkmark & {\color{red}23} & 24\checkmark \\ & & & & +4 \\ 25\checkmark & 26\checkmark & 27\checkmark & 28\checkmark \\ & & & & +4 \\ 29\checkmark & 30\checkmark & 31\checkmark & 32\checkmark \\ & & & & +4 \\ 33\checkmark & 34\checkmark & 35\checkmark & 36\checkmark \\ & & & & +4 \\ 37\checkmark & 38\checkmark & 29\checkmark & 40\checkmark \\ & & & & +4 \\ \dots\checkmark & \dots\checkmark & \dots\checkmark & \dots\checkmark \\ \hline \end{array}$

the largest number of cents that Larry cannot make exactly from a combination of 4- and/or 9-cent stamps is 23.