Modular Arithmetic: Fermat’s Theorem,
means:
equivalently:
for some integer
General Mod Arithmetic rules:
If
and
then
,
1. Mod 5:
Conclusion:
(mod
for all integers
multiple of 5.
2. Mod 7:
Conclusion:
(mod
for all integers
multiple of 7.
3. Fermat’s Little Theorem:
If
is a prime number and
is a number not divisible by , then
Equivalently,
(mod
.
(mod
.
Combinatorial Proof:
Given as many beads as you want in available colours, how many different types of
necklaces of beads can you form which use more than just 1 colour? is prime.
4. Calculate
(mod 11)
(mod 13)
(mod 11).
(mod 31)
Products:
Mod 35:
If we know a number
(mod 5) and
(mod 7), then we can find
(mod 35) uniquely.
5. Use the results from exercises 1 and 2 to fill in the first two rows, and then use these to
calculate the entries in the last row.
(mod 5)
(mod 5)
(mod 5)
(mod 7)
(mod 7)
(mod 7)
(mod 35)
(mod 35)
(mod 35)
Hint: If
a (mod 5) and
b (mod 7) , then
equation to find (mod 7). That’s enough to determine
b (mod 7). Use this
(mod 35).
The Chinese Remainder Theorem:
If
and
are relatively prime (no common factors >1, or equivalently gcd(
(mod
) and
(mod ), are enough to uniquely determine
(mod
)=1 ), then
).
6. In ancient China generals counted soldiers remaining after a battle by lining them up in
rows of different lengths and calculating the total from these remainders using what we
now call the Chinese Remainder Theorem. If a general had 1200 soldiers at the start of a
battle and if at the end there were 3 left over when they lined up 5 at a time, 3 left over
when they lined up 6 at a time, 1 left over when they lined up 7 at a time, and none left over
when they lined 11 at a time, how many soldiers survived the battle?
7. Let
i)
be a number not divisible by 5, 7, 11, or 13. Prove the following:
(mod 35),
ii)
(mod 1001).
Hint: Use the Chinese Remainder Theorem together with Fermat’s Little Theorem.
If
then
is a product of distinct primes and if
(mod
is not divisible by any of these,
).
Powers:
8. a) If
(mod 5), prove the following:
i)
(mod 25),
b) If
ii)
(mod 125),
iii)
(mod 625).
is not a multiple of 5, prove the following:
i)
(mod 25),
ii)
(mod 125),
Hint: a) i) Use
these factors mod 5. ii) Use
iii)
(mod 625).
and calculate each of
b) Use a) and Fermat’s Little Theorem.
In general:
9. a) Let
i)
be a prime number. If
(mod
b) If
),
ii)
is a prime number and
(mod ), then:
(mod
),
iii)
).
is a number not divisible by , then
(mod
Hint: a) iii): Induction, using
(mod
.
.
Euler’s Theorem:
If
is the prime factorization of
we define Euler’s number of
by:
.
If
is not divisible by any of these primes, then
(mod ).
10. Exercise:
= how many positive numbers smaller than
are relatively prime with
11. Find the last 3 digits of
Hint: Use Euler’s theorem to reduce the exponents to smaller numbers, and then also the
binomial formula (for 11=10+1, 14=10+4) to deal with the remaining powers.