Chapter 4: Theory of Congruences 15 SECTION C Solving Linear Congruences By the end of this section you will be able to solve congruences determine the number of solutions find the multiplicative inverse C1 Definition of a Linear Congruence In algebra we have linear equations in one unknown x given by 2x 1 7 Solving this equation gives x 3 . A linear congruence is an equation of the form ax b mod n The solution of this linear congruence is the set of integers x which satisfies this: ax b mod n Why is the solution a set of integers rather than a unique integer? Remember ax b mod n means that ax b kn for some integer k because by Definition (4.1): a b mod n a b is a multiple of n or there exists k such that a b kn In mathematics we see if we can obtain a unique solution. How can we get a unique solution out of a linear congruence? If two solutions x0 and x1 satisfy the linear congruence ax b mod n and they are congruent modulo n, that is x0 x1 mod n , then we say these are the same solution and count them as one solution. For example x 3 and x 8 satisfy (*) 2 x 1 mod 5 because 2 3 6 1 mod 5 and 2 8 16 1 mod 5 . Hence integers 3 and 8 satisfy the linear congruence (*). They are the same solution and we count them as one solution not two. We obtain these solutions by trial and error by putting integers for x into the linear congruence 2 x 1 mod 5 . In the next example we develop a more systematic way of finding the integers x which satisfies (*). Example 10 Determine the integers x of the following linear congruence: 2 x 1 mod 5 Solution By the above Definition (4.1)| we know that 2x 1 5k where k is an integer Re-arranging this gives 5k 1 x 2 Remember x must be an integer. So what values of k can we use? Chapter 4: Theory of Congruences 16 Only the odd numbers because if we chose even then we get even plus 1 which does give a whole number after dividing by 2. Substituting k 1, 3, 5, , 1, 3, 5, gives 5k 1 5 1 15 1 25 1 5 1 15 1 25 1 x , , , , , , , 2 2 2 2 2 2 2 3, 8, 13, , 2, 7, 12, Clearly there are infinite integers which satisfy the given linear congruence 2 x 1 mod 5 We count all these solutions as one or the same solution because they are all congruent to each other modulo 5. We normally write this as just one solution which is the least nonnegative residue modulo 5: x 3 mod 5 Example 11 Solve the linear congruence: 2 x 1 mod 4 Solution By Definition (4.1): a b mod n a b is a multiple of n or there exists k such that a b kn we have 2x 1 4k . Transposing this equation to make x the subject yields 4k 1 x 2 4k 1 What integers x satisfy this x ? 2 4k 1 There are no integers x such that x because 4k 1 is always odd. This means that 2 there is no solution to the given linear congruence 2 x 1 mod 4 . C2 Number of solutions of a Linear Congruence This means that there are some linear congruences which have no solution. How do we know which congruences have a solution? The next proposition gives the condition for a solution. Proposition (4.12). The linear congruence ax b mod n has a solution g b where g gcd a, n . Note that in Example 10 above the gcd 2, 5 1 and of course 11 so the linear congruence 2 x 1 mod 5 has a solution. On the other hand in Example 11 2 x 1 mod 4 the gcd 2, 4 2 and 2 does not divide 1 so there are no solutions to this linear congruence. How do we prove this proposition (4.12)? Chapter 4: Theory of Congruences 17 Well we use Proposition (2.9) on page 34 of Elementary Number Theory by David Burton: The linear Diophantine equation ax by c has a solution g c where g gcd a, b . Proof. We have ax b mod n which means that there is an integer k such that ax b kn implies ax kn b Let g gcd a, n . Then by Proposition (2.9) we conclude that ax kn b has a solution g b which is our required result. ■ Next we show that the linear congruence ax b mod n of Proposition (4.12) has exactly g incongruent solutions. What does incongruent mean? Not congruent. For example 6 x 2 mod 4 has solutions x 1 mod 4 and x 3 mod 4 but 3 1 mod 4 [Not congruent] We say x 1 mod 4 and x 3 mod 4 are two incongruent solutions of the given linear congruence 6 x 2 mod 4 . Proposition (4.13). The linear congruence ax b mod n If g b where g gcd a, n then this linear congruence has exactly g incongruent solutions modulo n. How do we prove this result? Again we use Proposition (2.9) on page 34 of Elementary Number Theory by David Burton: If x0 and y0 are any particular solutions of ax by c then all the other solutions are given by b a x x0 t , y y0 t g g where t is an arbitrary integer. Proof. From ax b mod n we know there is an integer k such that ax kn ax n k b Let x0 be a particular solution to this then by Proposition (2.9) n x x0 t where t is an arbitrary integer g is also a solution. Substituting t 0, t 1, t 2, , t g 1 we have n n n n x x0 , x0 , x0 2 , x0 3 , , x0 g 1 g g g g Need to show that these are not congruent modulo n. How? (*) Chapter 4: Theory of Congruences 18 Use proof by contradiction. Suppose any two in the list (*) are congruent modulo n: n n x0 m2 x0 m1 mod n g g where 0 m1 m2 g 1 . By using Definition (4.1) from page 1 on this congruence: a b mod n a b is a multiple of n or there exists k such that a b kn This means there is an integer k1 1 such that n n x0 m2 x0 m1 k1n g g n k1n g m2 m1 m2 m1 k1 g m2 m1 k1 g We have m2 m1 k1 g m1 g g (because k1 1 ) and earlier we had m2 g 1 . This is a contradiction. Hence none of the congruences in the above list (*) are congruent to each other modulo n. This means that the list n n n n x x0 , x0 , x0 2 , x0 3 , , x0 g 1 g g g g are incongruent modulo n. n Any other solution x x0 t is congruent to one of these in the list modulo n. Why? g We use the Division Algorithm on integers t and g. The Division Algorithm which is on page 17 of Elementary Number Theory by Burton: Let a, b 0 be given. Then there are unique integers q and r such that a bq r 0r b Applying this on t and g means there are integers q and r such that: t qg r 0 r g n Substituting this into x x0 t gives g n n x x0 t x0 qg r mod n g g n x0 nq r mod n g n x0 r mod n g n Since 0 r g so this x x0 t is in the above list. g ■ Chapter 4: Theory of Congruences 19 C3 Solving Linear Congruences The list produced in the proof of the above Proposition (4.13) is used to find the g solutions of ax b mod n . Hence n n n n x x0 , x0 , x0 2 , x0 3 , , x0 g 1 g g g g are the g solutions of ax b mod n provided g b . Example 12 Find all the solutions of the linear congruence: 7 x 35 mod 70 Solution First we determine the gcd of 7 and 70: gcd 7, 70 7 What next? Need to check that 7 divides into 35. Since 7 5 35 we have 7 incongruent solutions to the given linear congruence. We can find the first solution by trail and error. Is there an obvious solution? x 5 mod 70 . How do we find the other six solutions? Using the list in the proof n n n n x x0 , x0 , x0 2 , x0 3 , , x0 g 1 mod n g g g g n 70 With x0 5 mod 70 , n 70 and g 7 we have 10 g 7 x 5, 5 10, 5 2 10 , 5 3 10 , , 5 7 110 mod 70 x 5, 15, 25, 35, 45, 55, 65 mod 70 You can check that each of these is a solution by substituting these into 7 x 35 mod 70 . Example 13 Find all the solutions of the linear congruence: 7 x 34 mod 70 Solution First gcd 7, 70 7 but 7 does not divide into 34. What does this mean in relation to solutions of 7 x 34 mod 70 ? There are no solutions because by Proposition (4.12) which says: The linear congruence ax b mod n has a solution g b where g gcd a, n . Hence no solutions to 7 x 34 mod 70 . Example 14 Find all the solutions of the linear congruence: 5x 34 mod 7 Chapter 4: Theory of Congruences 20 Solution The gcd of 5 and 7 is 1, that is gcd 5, 7 1 and 1 divides into 34. How many solutions do we have to the given linear congruence? One solution which means this linear congruence has a unique solution. How can we find this? Well 5x 34 mod 7 means that there is an integer k such that 34 7k 5 We need to substitute an integer for k which results in x being an integer. k 3 will do: 34 7 3 34 21 x 11 5 5 We have 11 4 mod 7 . Hence the least non-negative residue modulo 7 is 4: 5 x 34 7k x x 4 mod 7 Checking that this solution is correct: 5 4 20 6 34 mod 7 To solve the linear equation 6x 15 0 It will easier to divide through by 3 and solve 2x 5 0 . Can we divide through by a common factor for congruences? Need to be careful because we are dealing with integers. The next example demonstrates this. Example 15 Find all the solutions of the linear congruence: 6 x 15 mod 21 Solution The gcd 6, 21 3 and 3 15 so there are 3 incongruent solutions modulo 21. If you only have paper and pen then modulo 21 is too hard to work with. Can we convert this to a smaller modulus and work with that? Yes. From page 10 of the last section we have: n Proposition (4.8). If ac bc mod n then a b mod where g gcd c, n . g We can write 6 x 15 mod 21 as 3 2 x 5 3 mod 7 3 By Proposition (4.8) we have 2 x 5 mod 7 By inspection we have x 6 mod 7 . From this we have x 6 7k where k is an integer What are values of k? Since we only have 3 solutions so k 0, 1 and 2 . By substituting these values we have x 6, 13, 20 mod 21 Chapter 4: Theory of Congruences 21 These are the 3 incongruent solutions modulo 21. You can check these by substituting these into the given linear congruence 6 x 15 mod 21 . It is generally easier to divide through by the gcd a, n to find solutions of ax b mod n because then we are dealing with a smaller modulus. C4 Unique Solutions How many solutions does the general linear congruence ax b mod n have if gcd a, n 1 ? Unique solution because g is the number of solutions of ax b mod n provided g divides b. We can write this as a general result. Corollary (4.14). If gcd a, n 1 then the linear congruence ax b mod n has a unique solution modulo n. Proof. Applying the above Proposition (4.13) on page 17 with g 1 : Proposition (4.13). The linear congruence ax b mod n If g b where g gcd a, n then this linear congruence has exactly g incongruent solutions modulo n. ■ Example 16 Solve the linear congruence: 6 x 1 mod 13 Solution gcd 6, 13 1 so we have a unique solution modulo 13. The congruence 6 x 1 mod 13 means that 1 13k 6 x 1 13k x where k 6 We chose k so that x is an integer. Let k 5 then x 11 . Hence x 11 mod 13 1 and this is the inverse or 6. 6 Similarly the unique solution of the above linear congruence 6 x 1 mod 13 is In ordinary algebra when we have 6 x 1 which gives x x 11 mod 13 We call this x 11 mod 13 the (multiplicative) inverse of 6 modulo 13. We can write this as the general result. Chapter 4: Theory of Congruences 22 C5 Multiplicative Inverse Definition (4.15). If ax 1 mod n and gcd a, n 1 then the unique solution of this congruence is called the multiplicative inverse of a modulo n. Example 17 Determine the inverse of 3 modulo 14. Solution To find the inverse means we need to solve 3x 1 mod 14 . By inspection x 5 mod 14 Inverse of 3 modulo 14 is 5 modulo 14. Example 18 Determine the inverse of 3 modulo 15. Solution In this case we need to solve 3x 1 mod 15 . Note that the gcd 3, 15 3 but 3 does not divide into 1 so there are no solutions to this congruence 3x 1 mod 15 . Hence 3 modulo 15 has no inverse. SUMMARY (4.12) ax b mod n has a solution g b where g gcd a, n . The multiplicative inverse of a modulo n is the unique solution of ax 1 mod n provided gcd a, n 1 .