A report about wilsons theorem and chinese remainder theorem

1. REPORT
IN
MATH 830
(Number Theory)
By Maria Katrina P. Miranda
To Engr. Rem Laodenio

2. WILSON’S THEOREM AND
CHINESE REMAINDER THEOREM
Objectives
At the end of the lesson, 80% of the students are expected to:
1. Define Wilson’s Theorem and Chinese Remainder Theorem.
2. Explain the basic facts about the theorems.
3. Practice creating proofs from the theorems.
4. Solve problems related to the topics.
5. Collect ideas after the presentation.
6. Evaluate own works.

3. WILSON’S THEOREM AND
CHINESE REMAINDER THEOREM
Definition of Terms
Wilson’s Theorem – According to Wikipedia, in number theory, it
states that a natural number n>1 is a prime number if and only
if the product of all the positive integers less than n is one less
than a multiple of n. That is (using the notations of modular
arithmetic), one has that the factorial (n-1)!=1x2x3…x(n-1)satisfies
(n-1)! ≡ -1 (mod n).

4. WILSON’S THEOREM AND
CHINESE REMAINDER THEOREM
Definition of Terms
Chinese Remainder Theorem - Is a theorem of number theory
( from Wikipedia) which states that if one knows the remainders of
the Euclidian Division of an integer n by several integers, then one
can determine uniquely the remainder of the division of n by the
product of these integers, under the condition that the divisors are
pairwise co-prime.

5. WILSON’S THEOREM AND
CHINESE REMAINDER THEOREM
“ There are certain things whose number is unknown.
When divided by 3, the remainder is 2; when divided by
5, the remainder is 3; and when divided by 7, the
remainder is 2. What will be the number of things?”
- Sunzi

6. WILSON’S THEOREM AND
CHINESE REMAINDER THEOREM
Activity
Directions:
Guess the terminologies to be used in
the given phrases. Arrange it in proper
order.

7. WILSON’S THEOREM AND
CHINESE REMAINDER THEOREM
Activity
Directions:
Choose from these options,
THEOREM
EXAMPLE
PROOF
CHINESE REMAINDER THEOREM

8. WILSON’S THEOREM AND
CHINESE REMAINDER THEOREM
Activity

9. WILSON’S THEOREM AND
CHINESE REMAINDER THEOREM
Activity

10. WILSON’S THEOREM AND
CHINESE REMAINDER THEOREM
Activity

11. WILSON’S THEOREM AND
CHINESE REMAINDER THEOREM
Wilson’s Theorem
If p is a prime, then (p-1) ! ≡ -1 (mod p).
 Proof
Let us take p>3, dismissing p = 2 and p = 3. Considering the
linear congruence ax ≡1 (mod p), then gcd(a,p)= 1. There is
a unique integer a, with 1≤ a’ ≤ p-1. Satisfying aa’ ≡ 1(mod p).

12. WILSON’S THEOREM AND
CHINESE REMAINDER THEOREM
 Proof
Because p is prime, a=a’ if and only if a=1 or a=p-
1. The congruence a2 ≡ 1(mod p) is equivalent to
(a-1)(a+1)≡ 0(mod p). Therefore, either a-1 = 0
(mod p) and a=1 or a+1= 0 (mod p) and so a=p-1.

13. WILSON’S THEOREM AND
CHINESE REMAINDER THEOREM
 Proof
If we omit the numbers 1 and p-1, the effect is to
group the remaining integers 2,3,…p-2 into pairs a,
a’ where a ≠ a’ such that aa’ ≡ 1(mod p). When
these (p-3)/2 congruence are multiplied together
and the factors rearranged, we get 2*3 …(p-2) ≡ 1
(mod p).

14. WILSON’S THEOREM AND
CHINESE REMAINDER THEOREM
Proof
Or
(p-2)! ≡ 1 (mod p)
Now multiply by p-1 to obtain the congruence
(p-1) ≡ p-1 ≡ -1 (mod p)
As was to be proved. □

15. WILSON’S THEOREM AND
CHINESE REMAINDER THEOREM
Example
Let p = 13, then divide the integers
2,3,…,11 into (p-3)/2 = 5 pairs, each
product of which is congruent to 1 modulo
13.

16. WILSON’S THEOREM AND
CHINESE REMAINDER THEOREM
Example
2*7 ≡ 1 (mod 13)
3*9 ≡ 1 (mod 13)
4*10 ≡ 1 (mod 13)
5*8 ≡ 1 ( mod 13)
6*11 ≡ 1 (mod 13)

17. WILSON’S THEOREM AND
CHINESE REMAINDER THEOREM
Example
Multiplying,
n! = (2*7)(3*9)(4*10)(5*8)(6*11)
≡ 1 (mod 13)
12! ≡ 12 ≡ -1 (mod 13)
Thus, (p-1)! ≡ -1(mod p), with p = 13. ▄

18. WILSON’S THEOREM AND
CHINESE REMAINDER THEOREM
Chinese Remainder Theorem
Let a and b be natural numbers with gcd(a,b)
= 1, and let c and d be arbitrary integers.
Then, there is a solution to the simultaneous
congruence.

19. WILSON’S THEOREM AND
CHINESE REMAINDER THEOREM
 Chinese Remainder Theorem
x ≡ c mod a
x ≡ d mod b
Moreover, the solution is unique modulo ab; that is,
if X1 and X2 are two solutions, then X1 ≡ X2 mod
ab.

20. WILSON’S THEOREM AND
CHINESE REMAINDER THEOREM
Proof
Since gcd(a,b) = 1, there are integers u and v
with ua+bv=1.
Now let
x ≡ dau + cbv.

21. WILSON’S THEOREM AND
CHINESE REMAINDER THEOREM
 Proof
We have bv ≡ 1 mod a; and av ≡ 1 mod b. So, x ≡ dav
≡ d mod b.
If X1 and X2 are two solutions, then X1 ≡ c ≡ X2 mod a
and X1 ≡ d ≡ X2 mod b. Since a and b are co prime, ab
divides X1 - X2 , so that X1 ≡ X2 mod ab.

22. WILSON’S THEOREM AND
CHINESE REMAINDER THEOREM
Example
Find all numbers congruent to 2 mod 3, 1 mod
4 and 3 mod 5.
-3+4 = 1
-3*1+4*2 = 5.

23. WILSON’S THEOREM AND
CHINESE REMAINDER THEOREM
Example
Look for a number congruent to 5 mod 12 and
3 mod 5.
-2*12+5*5 = 1
-2*12*3+5*5*5 = 53

24. WILSON’S THEOREM AND
CHINESE REMAINDER THEOREM
Example
Hence,
the general solution is the congruence
class [53] 60 (all numbers congruent to 53 mod
60).

25. WILSON’S THEOREM AND
CHINESE REMAINDER THEOREM
Evaluation:
Rank your answers from the scale of 1 to 5 where 5 is the highest.
1. Are you satisfied with the content of this
course/topic?
2. Do you find this topic/course applicable to your
field.

26. WILSON’S THEOREM AND
CHINESE REMAINDER THEOREM
 Evaluation:
3. Are you able to understand your instructor’s explanation?
4. Did you find the course interesting?
5. On the scale of 1 to 5, what is your general feeling
about the course?

27. WILSON’S THEOREM AND
CHINESE REMAINDER THEOREM
Sources:
• Elementary Number Theory by David M. Burton
• Introduction to Number Theory and Its Applications by
Lucia Moura
• Instructional Design: Using the Addie Model to build a
writing course for University students by R.T. Williams
• Wikipedia

28. WILSON’S THEOREM AND
CHINESE REMAINDER THEOREM
Thank You!!!