Remainder on division

One function that appears to be missing from the fx-50F Plus is finding the remainder when one integer is divided by another. For example, what is the remainder when 4593 is divided by 7, or putting this another way, what is 4593 congruent to modulo 7? The fx-50F does have the ability to calculate fractions and if you key in 4593 [a b/c] 7 [EXE] you obtain 656_1_7 which shows that 4593=656x7+1, that is the remainder when 4593 is divided by 7 is 1. Equivalently 4593 is congruent to 1 modulo 7. You have to be wary that if your divisor is not prime, then you may have to adjust the numerator of your fraction accordingly to obtain the correct remainder. For example, if you enter 4593 [a b/c] 21 [EXE] you obtain 218_5_7. Here the remainder is not 5 but 5x3=15 since the denominator is 7, not 21, in the resulting fraction. You need to multiply the fraction 5/7 top and bottom by 3 to get the correct result, that is 4593=218x21+15.

An alternative method is, of course, to carry out the division in decimals 4593/7=656.1428571. If you then take this number, subtract 656 and multiply by 7 you obtain remainder 1. This seems simple enough. But what happens if you want to do division on very large numbers? It all starts to get a bit messy and inaccurate, partly because of the number of digits that the fx-50F can accurately hold and partly because only ten digits are displayed on the screen.

For example, Edouard Lucas had shown in 1876 that the number 2⁶⁷-1 =147,573,952,589,676,412,927 is not prime but no one knew what its factors were. In 1903 Frank Nelson Cole (1861-1926) famously gave a presentation to the American Mathematical Society where he proceeded to calculate, in silence, 2⁶⁷-1 on side of the blackboard and then, on the other, the result, by hand, of 193,707,721 × 761,838,257,287. Cole, having demonstrated that the two numbers were equal, sat down, not uttering a word. He received a standing ovation!

What I would like to do is to demonstrate that we can show that either of these factors will divide this number by creating a program that will calculate that the remainder on division is zero. Furthermore, this program will cope with a natural number of any length (though, the divisor will have to less than the largest positive integer that the calculator can accurately hold).