Final answer:
Mathematical proofs show that creating even postage with even numbers of both stamp types always results in an even total, an even postage amount necessitates an even number of at least one stamp type, and making 72 cents in postage requires at least 6 of one type of stamp due to the nature of the numbers involved.
Step-by-step explanation:
Mathematical Stamp Problems
Addressing the provided questions one by one:
- To prove that using an even number of both 5-cent and 8-cent stamps results in an even amount of postage. If we have even numbers of both stamps, let's say 2n 5-cent stamps and 2m 8-cent stamps, their total value will be 10n + 16m cents, which is clearly an even number since the sum or product of even numbers is always even.
- To prove that an even amount of postage means an even number of at least one type of stamp: Assume you have an odd number of 5-cent stamps (5(2n+1)) and an odd number of 8-cent stamps (8(2m+1)). The sum would be 10n+5+16m+8, an odd number plus an odd number, which is even; however, to have an even sum starting with even postage requires that one of these initial components must also be even.
- Finally, to prove that making exactly 72 cents of postage requires at least 6 of one type of stamp: Since 5 and 8 have a common factor of 1, we can use Bezout's identity. However, trying to write 72 as 5x + 8y, where x and y are integers, shows that getting a sum of 72 requires multiples that go beyond small numbers. For instance, using 9 8-cent stamps gives us 72 cents, fulfilling the condition.
The three parts emphasize the use of modular arithmetic and linear combinations in problem-solving.