ASSIGNMENT 4  MATH235, FALL 2007
Submit by 16:00, Tuesday, October 9 (use the designated mailbox in Burnside Hall,
10
th
floor).
1. Let
a
=
p
r
1
1
p
r
2
2
· · ·
p
r
k
k
and
b
=
p
s
1
1
p
s
2
2
· · ·
p
s
k
k
, where
p
1
, p
2
, . . . , p
k
are distinct positive primes and each
r
i
, s
i
≥
0. Using unique factorization, prove that
(1)
(
a, b
) =
p
n
1
1
p
n
2
2
· · ·
p
n
k
k
, where
n
i
= min(
r
i
, s
i
).
(2)
[
a, b
] =
p
t
1
1
p
t
2
2
· · ·
p
t
k
k
, where
t
i
= max(
r
i
, s
i
).
2. The least common multiple of nonzero integers
a, b
is the smallest positive integer
m
such that
a

m
and
b

m
. We denote it by lcm(
a, b
) or [
a, b
]. Prove that:
(1)
If
a

k
and
b

k
then [
a, b
]

k
.
(2)
[
a, b
] =
ab
(
a,b
)
if
a >
0
, b >
0.
3. Prove or disprove: If
n
is an integer and
n >
2, then there exists a prime
p
such that
n < p < n
!.
4. Find all the primes between 1 and 150. The solution should consist of a list of all the primes + giving
the last prime used to sieve + explanation why you didn’t have to sieve by larger primes.
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 Fall '07
 Goren
 Math, Algebra, Prime number, 1 2 k, nk nk1, 1 2 1 2 k

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