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Moreover, it would give us much more information than we need.Since we want only the tens and units digits of the number in question, it suffices to find the remainder when the number is divided by . By the "multiplication" property above, then, it follows that (mod ).
We begin by writing down the first few powers of mod : A pattern emerges! Since both integers are positive, this means that they share the same tens and units digits. Can you find a number that is both a multiple of but not a multiple of and a perfect square? Rewriting the question, we see that it asks us to find an integer that satisfies . Now, all we are missing is proof that no matter what is, will never be a multiple of plus , so we work with cases: This assures us that it is impossible to find such a number.We can rewrite each of the integers in terms of multiples of and remainders: .When we add all four integers, we get And as we did in the problem above, we can apply more pairs of equivalent integers to both sides, just repeating this simple principle. Adding the two equations we get: Which is equivalent to saying The same shortcut that works with addition of remainders works also with subtraction. Modular arithmetic provides an even larger advantage when multiplying than when adding or subtracting.Let's use a clock as an example, except let's replace the at the top of the clock with a .Starting at noon, the hour hand points in order to the following: This is the way in which we count in modulo 12. The same is true in any other modulus (modular arithmetic system).In fact, the advantage in computation is even larger and we explore it a great deal more in the intermediate modular arithmetic article. After that, we see that 7 is congruent to -1 in mod 4, so we can use this fact to replace the 7s with -1s, because 7 has a pattern of repetitive period 4 for the units digit.Note to everybody: Exponentiation is very useful as in the following problem: What is the last digit of if there are 1000 7s as exponents and only one 7 in the middle? is simply 1, so therefore , which really is the last digit. We could (in theory) solve this problem by trying to compute , but this would be extremely time-consuming.Modular arithmetic is a special type of arithmetic that involves only integers.This goal of this article is to explain the basics of modular arithmetic while presenting a progression of more difficult and more interesting problems that are easily solved using modular arithmetic.Free online quizzes on Math Word Problems and Math Exercises including Addition, Subtraction, Multiplication, Division and Arithmetic problems. See also, Kids Math Word Problems II for kids 8-9 years of age.