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URGENT! I NEED HELP RIGHT AWAY!

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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.

What is the largest number of cents that Larry CANNOT make exactly from a combination of 4- and/or 9-cent stamps?

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.)

Jan 26, 2018

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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.

What is the largest number of cents that Larry CANNOT make exactly from a combination of 4- and/or 9-cent stamps?

Explain how you know your answer is correct. (You should explain two things:
why he CAN make any bigger amount.)

$$\begin{array}{l} \cdots\{4\} \cdots \{8,9\}\cdots \{12,13\}\cdots \{16,17,18\} \cdots \{20,21,22\} \cdots {\color{red}23} \{\underbrace{24,25,26,27}_{\underset{27+4=31}{\underset{26+4=30,}{\underset{25+4=29,}{\underset{24+4=28,}{}}}} }\} \{\underbrace{28,29,30,31}_{\underset{31+4=35}{\underset{30+4=34,}{\underset{29+4=23,}{\underset{28+4=32,}{}}}} }\} \{\underbrace{32,33,34,35}_{\underset{35+4=39}{\underset{34+4=38,}{\underset{33+4=37,}{\underset{32+4=36,}{}}}} }\} \text{ and so on} \\\\ \{\text{reachable}\} \\ \cdots \text{not reachable}\\ \end{array}$$

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

After four consecutive reachable numbers, 24, 25, 26, 27, he CAN make any bigger amount. Jan 26, 2018