Difference between revisions of "Modular arithmetic"
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Revision as of 16:39, 28 June 2006
Modular arithmetic is a special type of arithmetic that involves only integers. Since modular arithmetic is such a broadly useful tool in number theory, we divide its explanations into several levels:
Contents
[hide]Resources
Introductory Resources
- The AoPS Introduction to Number Theory by Mathew Crawford.
The Natural Appeal of Modular Arithmetic
Observe that we use modular arithmetic even when solving some of the most basic, everyday problems. For example:
Cody is cramming for an exam that will be held at 2 PM. It is the morning of the day of the exam, and Cody did not get any sleep during the night. He knows that it will take him exactly one hour to get to school from the time he wakes up, and he insists upon getting at least five hours of sleep. At what time in the morning should Cody stop studying and go to sleep?
We know that the hours of the day are numbered from to , with hours having the same number if and only if they are a multiple of hours apart. So we can use subtraction mod to answer this question.
We know that since Cody needs five hours of sleep plus one hour to get to school, he must stop studying six hours before the exam. We can find out what time this is by performing the subtraction
So Cody must quit studying at 8 AM.
Of course, we are able to perform calculations like this routinely without a formal understanding of modular arithmetic. One reason for this is that the way we keep time gives us a natural model for addition and subtraction in : a "number circle." Just as we model addition and subtraction by moving along a number line, we can model addition and subtraction mod by moving along the circumference of a circle. Even though most of us never learn about modular arithmetic in school, we master this computational model at a very early age.
A Word of Caution
Because of the way we define operations in , it is important to check that these operations are well-defined. This is because each of the sets that make up contains many different numbers, and therefore has many different names. For example, observe that in , we have and . It is reasonable to expect that if we perform the addition , we should get the same answer as if we compute , since we are simply using different names for the same objects. Indeed, the first addition yields the sum , which is the same as the result of the second addition.
The "Useful Facts" above are the key to understanding why our operations yield the same results even when we use different names for the same sets. The task of checking that an operation or function is well-defined, is one of the most important basic techniques in abstract algebra.
Computation of Powers Mod n
The "exponentiation" property given above allows us to perform rapid calculations modulo . Consider, for example, the problem
What are the tens and units digits of ?
We could (in theory) solve this problem by trying to compute , but this would be extremely time-consuming. 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 . In other words, all of the information we need can be found using arithmetic mod .
We begin by writing down the first few powers of :
A pattern emerges! We see that (mod ). So for any positive integer , we have (mod ). In particular, we can write
(mod ).
By the "multiplication" property above, then, it follows that
(mod ).
Therefore, by the definition of congruence, differs from by a multiple of . Since both integers are positive, this means that they share the same tens and units digits. Those digits are and , respectively.
A General Algorithm
In the example above, we were fortunate to find a power of -- namely, -- that is congruent to mod . What if we aren't this lucky? Suppose we want to solve the following problem:
What are the tens and units digits of ?
Again, we will solve this problem by computing modulo . The first few powers of are
This time, no pattern jumps out at us. Is there a way we can find the power of without taking this list all the way out to the term -- or even without patiently waiting for the list to yield a pattern?
Suppose we condense the list we started above; and instead of writing down all powers of , we write only the powers , where is a power of . We have
(Observe that this process yields a pattern of its own, if we carry it out far enough!)
Now, observe that, like any positive integer, can be written as a sum of powers of two:
We can now use this powers-of-two expansion to compute :
So the tens and units digits of are and , respectively.
We can use this method to compute modulo , for any integers and , with . The beauty of this algorithm is that the process takes, at most, approximately steps -- at most steps to compute the values for a power of two less than , and at most steps to multiply the appropriate powers of according to the binary representation of .
This method can be further refined using Euler's Totient Theorem.