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1. Theorem: That there exist natural numbers n and m that fulfill the
equation 3m + 5n = 12.
Since n is a natural number, only natural multiples of 5 are relevant. This
leaves us with two possible options for 5n: 5 (if n = 1) and 10 (if n = 2), as
0 < 5n ≤ 12.
The equation can be written as 3m = 12 - 5n for simplicity.
The difference between 12 and 5 is 7, and ¬(3|7). Therefore, 5n cannot
be 5 and n cannot be 1.
The other option is for n = 2. 12 - 10 = 2, and ¬(3|2). Therefore, 5n
cannot be 10 and n cannot be 2.
There are no more options for n, and so(∃m∈N)(∃n∈N)(3m + 5n = 12)
is false.
proof1.pdf (PDF, 69.34 KB)
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