AwesomeMath 2007
Track 4 — Modulo Arithmetic
Week 2
Lecture 6 : Divisibility and the Euclidean Algorithm
Yufei Zhao
July 24, 2007
1. If
a
and
b
are relatively prime integers, show that
ab
and
a
+
b
are also relatively prime.
2.
(a) If 2
n
+ 1 is prime for some integer
n
, show that
n
is a power of 2.
(b) If 2
n

1 is prime for some integer
n
, show that
n
is a prime.
3. Show that the fraction
12
n
+ 1
30
n
+ 2
is irreducible for all positive integers
n
.
4. Let
x, a, b
be positive integers, show that gcd(
x
a

1
, x
b

1) =
x
gcd(
a,b
)

1.
5.
(a) Let
p
be a prime number. Determine the greatest power of
p
that divides
n
!, where
n
is
a positive integer.
(b) Let
m
and
n
be positive integers. Show that
(
m
+
n
)!
m
!
n
!
is an integer (without referring
to binomial coefficients).
6. (USAMO 1972) Show that
gcd(
a, b, c
)
2
gcd(
a, b
) gcd(
b, c
) gcd(
c, a
)
=
lcm(
a, b, c
)
2
lcm(
a, b
) lcm(
b, c
) lcm(
c, a
)
.
7.
(a) Show that if
a
and
b
are relatively prime integers, then gcd(
a
+
b, a
2

ab
+
b
2
) = 1 or 3.
(b) Show that if
a
and
b
are relatively prime integers, and
p
is an odd prime, then
gcd
a
+
b,
a
p
+
b
p
a
+
b
= 1 or
p.
8. Let
n
be a positive integer.
(a) Find
n
consecutive composite numbers.
(b) Find
n
consecutive positive integers, none of which is a power of a prime.
9. Let
n >
1 be a positive integer. Show that
1 +
1
2
+
1
3
+
· · ·
+
1
n
is not an integer.
(Try not to use any powerful results about the distribution of prime
numbers.)
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