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SW-MO ARML
Practice – September 20, 2008
Modulus
You have actually dealt with modulus since elementary school – every time you worked
clock arithmetic problems. For example:
Problem: If you start to work at 10:00 am and work for 5
hours, at what time do you finish?
Solution: 10  5  3 mod 12 You finish at 3 pm.
Problem: If class begins at 9:15 and ends at 10:02, for
how many minutes are you in class?
Solution: 2  15  13 mod 60  47 mod 60 so you are in
class for 47 minutes.
WHAT IS MODULUS?
If a  b(mod m) then a – b is evenly divisible by m. Another way of thinking of it is that
a is the remainder when b is divided by m. Try some basic problems:
1. 10(mod 7)  _____
2. 20(mod 7)  _____
3. 18(mod 7)  _____
4.  19(mod 5)  _____
5. 19(mod 5)  _____
SOME USEFUL PROPERTIES
For each of the following properties of modulus, give an example using numbers to help
yourself feel more comfortable with what it is saying.
Let a  a (mod m) and b  b (mod m) , then the following properties hold:
1. Equivalence: If a  b(mod 0) , then a = b. (Consider this a definition. WHY?)
2. Determination: Either a  b(mod m) or a is not  b( mod m) .
3. Reflexivity: a  a(mod m)
4. Symmetry: If a  b(mod m) then b  a(mod m)
5. Transitivity: If a  b(mod m) and b  c(mod m) , then a  c(mod m)
6. a  b  a   b(mod m)
7. a  b  a   b(mod m)
8. a  b  a   b(mod m)
9. If a  b(mod m) then ka  kb(mod m)
10. If a  b(mod m) , then a n  b n (mod m)
11. If a  b(mod m) and a  b(mod n) , then a  b(mod[ m, n]) where [m,n] is the
LCM of m and n.

m 
 where (k,m) is the GCF of k and m.
12. If ak  bk (mod m) , then a  b mod
k , m 

13. If a  b(mod m) , then P(a)  P(b)(mod m) , for P(x) any polynomial.
SOME RELATED PROBLEMS
1. Find the remainder when 5 2008 is divided by 6.
 x  1(mod 2)
2. Give three solutions in the natural numbers to the system: 
3x  2(mod 5)
3. What can you conclude about a if a  0(mod m) ?
4. How many integers are there in the interval [50, 250] which are congruent to
1(mod7)?
5. Prove that if the sum of the divisors of a number is divisible by three, then the
number is divisible by three. (Use mod 3.)
6. What is the divisibility test for 11? Can you verify it using mod11?