Download 1. On Repunits. A repunit is a positive integer all of whose digits are

1.
On Repunits. A repunit is a positive integer all of whose digits are 1. Let Ω n
denote the repunit with n digits—thus, for example, Ω 2 = 11 , and Ω 3 = 111 .
c
Find the quotient and the remainder for any n when one performs the
division algorithm with dividend Ω n and divisor Ω 2 .
d
Find the quotient and the remainder for any n when one performs the
division algorithm with dividend Ω n and divisor Ω 3 .
2.
Eric and Laura start jogging from the same place at the same time on a circular
track. Laura, the faster one, circles the track every five minutes while Eric takes 8
minutes to do so. They will keep jogging that way until they meet again at the
start.
c
How long will they jog?
d
Including the starting point, in how many different places along the track
will they meet?
3.
On the Euclidean Algorithm.
c
Find the g.c.d. of 2736734 and 3259256 by doing the Euclidean
Algorithm.
d
Perform the algorithm using the round-off quotients rather than the rounddown ones.
e
Write the g.c.d. as a linear combination of 2736734 and 3259256.
4.
On a Different Algorithm. The ancient Egyptians has a way of multiplying any
two numbers, but never doing a multiplication by anything bigger tahn 2. Simialrly, there
is a way of performing the Euclidean algorithm, and never having to divide by anything
bigger than 2. Here is the process.
Suppose we will peeform the algorithm to a and b. It takes 3 columns. Define a 0 = a ,
b0 = b , and c0 = 1 .
If a n = bn , stop, and the g.c.d. is given by a n c n .
If a n and bn are both even, then a n +1 =
an
2
, bn +1 =
bn
2
and c n +1 = 2c n .
If a n and bn are both odd, then a n+1 = a n − bn , bn +1 = min(a n , bn ) and c n +1 = c n .
c
d
Apply this new algorithm to 59059 qand 12508.
Make a simple conjecture about the behaviour of c thorugh the algorithm
and prove it.
If a n and bn are of different parity, then the even one gets divided by 2, the odd remains
the same, and so does c.
5.
For which m does 552 x + 795 y = m have a solution?
6.
7.
Consider the line y = αx where α is a real number. Find necessary and sufficient
conditions on α for the line to contain a lattice point.
Consider the sequence of remainders when the Euclidean algorithm is applied to
r0 and r1 where r0 > r1 > 0 .
c
d
rm
for all appropriate m’s;
2
Give a reasonable upper bound for the number of steps that the Euclidean
algorithm will take if r0 has a 1000 digits.
Prove rm+ 2 <
8.
As usual, let Fn denote the nth term of the Fibonacci sequence: 1,1,2,3,5,8,… .
Apply the Euclidean algorithm to two consecutive terms. What is the sequence of
quotients? Describe precisely how many terms the sequence of quotients has and
what they are.
9.
On Legs of Pythagorean Triples.
c
Prove every odd positive integer is the numerator of a reduced rational
point of the unit circle.
d
Which integers are possible for the numerator of a reduced rational point
of the unit circle?
e
Which integers are possible for the numerator of a rational point of the
unit circle?
Fill in as many of the following statements as you can (by ends we mean the last
digit):
7 n ends in
if and only if n ≡
mod
n
7 ends in
if and only if n ≡
mod
n
7 ends in
if and only if n ≡
mod
n
7 ends in
if and only if n ≡
mod
n
7 ends in
if and only if n ≡
mod
10.
11.
12.
Find a divisor of x 20 + 1 .
n
Let Fn = 2 2 + 1 denote the nth Fermat number. Show that if n < m , then Fn and
Fm have no common divisor. Hint: Use clocks.
13.
As wonderful as Theorem 3 is, it has its limitations. One subtle error that one is
prompt to commit is the following false conclusion from it:
if s ≡ t mod n, then a s ≡ a t mod n.
c
Find s ≡ t mod 7 such that 2 s ≡/ 2 t mod 7.
d
Find an n such that s ≡ t mod n implies 2 s ≡/ 2 t mod 7.
e
Do part d with 7 replaced by 13.
14.
Find the remainder of 105,837,263,938,943 when divided by each of 2, 3, 4, 5, 6,
8, 9, 10, 11, 12, 15, 24 and 25. Give brief reasons for your answers or circle the
crucial information.
15.
Use the squaring algorithm to do each of the following computations:
c
The last digit of 2 123 .
d
The last two digits of 3 20 .
e
f
g
The last two digits of 7 30 .
The last three digits of 1140 .
The last three digits of 2 100 .