DIvisibility Error in the TI-84

Question

I am trying to factorize a large number,

$$600851475143$$

So I just started trying primes and in my TI-84, I got that

$$600851475143=61\cdot 9850024183$$

so then I got

$$9850024183=7\cdot 1407146312$$

Which can't be possible since this implies the original number is even as a factor is even itself. When checking other sources, it is clear $61$ is NOT a prime factor. But why then would my calculator give an incorrect factor to begin with?

Answer 1

Often cases such as this are caused by floating point error, which is worth being familiar with. In this case, however, it is highly unlikely that floating point error is at all relevant, given the scale of these computations. What seems to be more likely is that you have a display precision setting active on your calculator, which is causing you calculator to round answer before displaying them. You may want to navigate to the MODE menu and check which setting is highlighted on the third line that reads FLOAT 1 2 3 4 5 6 7 8 9.

Correction: the TI-84 series can only display 10 significant figures regardless of display settings. You may wish to use a different device to perform these computations, such as a computer algebra system like Sage, in order to make use of greater precision. This has the added benefit of allowing you to automate many common tasks. For instance, you can trivially factor any integer in sage with the following:

sage: factor(600851475143)
71 * 839 * 1471 * 6857

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As mentioned in the comments, there is also the remainder function which can be found under MATH>NUM>0:remainder( which can be used to more effectively test the divisibility of two integers.

Answer 2

long division saves the day... BTW that's tiny compared to what you can attempt once you know a bit of theory. 10 digits precision as suggested in the other answer, is also a low amount. On to the long division (using the theory you can changes things by a multiple and have the same remainder):$$\begin{align}9850024182\\61\overline{\big)600851475143}\\\underline{-549}\phantom{000000000}\\518\phantom{00000000}\\\underline{-488}\phantom{00000000}\\305\phantom{0000000}\\\underline{-305}\phantom{0000000}\\0147\phantom{0000}\\\underline{-0122}\phantom{0000}\\255\phantom{000}\\\underline{-244}\phantom{000}\\111\phantom{00}\\\underline{-61}\phantom{00}\\504\phantom{0}\\\underline{-488}\phantom{0}\\163\\\underline{-122}\\41\end{align}$$ which shows it doesn't divide evenly in the first case. ( just a pain to align,without a rewrite)

Emphasis A calculator is only as good generally, as it's operator. some calculators don't have order of operations built in for example, you should, it's best left as a checking mechanism in most cases.