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30
(iv)
For all rationals
a
,
b
,
c
, prove
δ
p
(
a
,
b
)
≤
δ
p
(
a
,
c
)
+
δ
p
(
c
,
b
)
.
Solution.
δ
p
(
a
,
b
)
≤
δ
p
(
a
,
c
)
+
δ
p
(
c
,
b
)
because
δ
p
(
a
,
b
)
=k
a
−
b
k
p
=k
(
a
−
c
)
+
(
c
−
b
)
k
p
≤
max
{k
a
−
c
k
p
,
k
c
−
b
k
p
}
k
a
−
c
k
p
+k
c
−
b
k
p
=
δ
p
(
a
,
c
)
+
δ
p
(
c
,
b
).
(v)
If
a
and
b
are integers and
p
n

(
a
−
b
)
, then
δ
p
(
a
,
b
)
≤
p
−
n
.
(Thus,
a
and
b
are
“
close
”
if
a
−
b
is divisible by a
“
large
”
power
of
p
.)
Solution.
If
p
n

a
−
b
, then
a
−
b
=
p
n
u
, where
u
is an integer.
But
k
u
k
p
≤
1 for every integer
u
, so that
δ
(
a
,
b
)
=k
a
−
b
k
p
=k
p
n
u
k
p
=k
p
n
k
p
k
u
k
p
≤
p
−
n
.
At this point, one could assign a project involving completions,
p
adic integers, and
p
adic numbers.
1.74
Let
a
and
b
be in
Z
. Prove that if
δ
p
(
a
,
b
)
≤
p
−
n
, then
a
and
b
have the
same
f
rst
np
adic digits,
d
0
,...,
d
n
−
1
.
Solution.
This follows from the fact that
p
n

a
if and only if the
f
rst
p
adic digits
d
0
,...,
d
n
−
1
are all 0.
1.75
Prove that an integer
M
≥
0 is the lcm of
a
1
,
a
2
,...,
a
n
if and only if it
is a common multiple of
a
1
,
a
2
,...,
a
n
which divides every other common
multiple.
Solution.
Consider the set
I
of all the common multiples of
a
1
,
a
2
,...
,
a
n
.
It is easy to check that
I
satis
f
es the hypotheses of Corollary 1.37, so that
every number
m
in
I
is a multiple of
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This note was uploaded on 12/21/2011 for the course MAS 4301 taught by Professor Keithcornell during the Fall '11 term at UNF.
 Fall '11
 KeithCornell

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