diophantine equations - How does modular arithmetic work - Fermat's last theorem near misses?

This question about Fermat's Last Theorem near misses uses modular arithmetic (converting all the terms to mod 100) to show why $$3987^{12}+4365^{12}\neq4472^{12}.$$

I've only just come across modular arithmetic, and the method looks really neat. But I'm puzzled as to why this works. In other words, I think I'm asking if an equation is true in modular form, how do we know it's true in ordinary integer form? Apologies in advance if this is a really dumb question.

Answer

Since you're new to modular arithmetic, here is a very simple explanation using a couple of examples.

There are three types of modular arithmetic you are already familiar with from grade school: mod 10, mod 5, and mod 2.

Mod 2 just refers to even numbers and odd numbers. Even numbers are those which are "equal to" (actually "congruent to") 0 (mod 2). Odd numbers are those which are congruent to 1 (mod 2).

For mod 5, discard all but the last digit of a number, then (if it is greater than 4), subtract 5 from the remaining digit.

For mod 10, just take the last digit of the number.

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Consider the following equation. Is it true?

$5723784602 + 2893649283 = 8617433887$

Using modular arithmetic (specifically, mod 10) you can discard all but the final digit of each number, and state:

$2 + 3 \equiv 7 \mod 10$

This is obviously false. Therefore, the original equation is false.

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How about the following equation?

$234343 \times 23845 = 5587908832$

Using the rules that you were probably taught in Grade School, if you take any number that ends in a five and multiply it by anything, the product must end in either a five or a zero. Therefore, this is false.

We can state this with modular arithmetic as follows:

$3 \times 0 \equiv 2 \mod 5$

Obviously this is false. Anything times zero must equal zero.

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However, the reverse approach doesn't work:

$29834620934 + 293840239843 = 17$

If we check this with modular arithmetic (mod 10), we get:

$4 + 3 \equiv 7 \mod 10$

This is true, but the original equation is false.

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In summary: You can use modular arithmetic to prove an equation false.

You can't use it (in this simplistic form) to prove an equation true.