You\'re on a remote island hoping to send some postcards and letters which requi

Question

You're on a remote island hoping to send some postcards and letters which require various amounts of postage. The post office only sells 4-cent and 5-cent stamps, but the woman at the counter claims it's not a problem. She says: "Every amount of postage that is at least 12 cents can be made from a combination of 4-cent stamps and 5-cent stamps." For example, she says, if the postage is 13 cents, then you can make it using 1 5-cent stamp and 2 4-cent stamps. Play around with some other small examples and then decide if she is correct or not (in other words, decide if the statement is true or false). If the statement is true, prove it. If it is false, "salvage it" by making some changes so you have a new true statement and then prove your new statement.

Explanation / Answer

4m+5n=A, or that A is the sum of two numbers either of which is divisible by 4 or 5. ; A an element of the natural numbers >= 12. Base case: 12; when m=3 and n=0. Now assuming 4m+5n=A Prove 4j+5k=A+1 Let j=m-1 and k=n+1, WOLOG, which is allowed since j,m,k,n are all arbitrary natural numbers. 4(m-1)+5(n+1)=A+1 4m-4+5n+5=A+1 4m+5n=A; A+1=A+1, which is a tautology.

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