Solutions, Algebra hw, Week 2
by Bracket
0.6.
For driver’s license numbers issued in New York prior to September of 1992, the three digits preceding
the last two of the number of a male with birth month
m
and birth date
b
are represented by 63
m
+ 2
b
.
For females the digits are 63
m
+ 2
b
+ 1. Determine the dates of birth and sex(es) corresponding to the
numbers 248 and 601.
248
: 248 = 63
·
3 + 2
·
29 + 1
March 29, Female
601
: 601 = 63
·
9 + 2
·
17 + 0
September 17, Male
0.13.
Let
n
and
a
be positive integers and let
d
= gcd(
a, n
). Show that the equation
ax
mod
n
= 1 has a
solution if and only if
d
= 1.
(From Gallian) Suppose that there is an integer
n
such that
ab
mod
n
= 1. Then there is an integer
q
such that
ab

nq
= 1. Since
d
divides both
a
and
n
,
d
also divides 1. So,
d
= 1. On the other
hand, if
d
= 1, then by Theorem 0.2, there are integers
s
and
t
such that
as
+
nt
= 1. Thus, modulo
n
,
as
= 1.
0.16.
Use the Euclidean Algorithm to find gcd(34
,
126) and write it as a linear combination of 34 and 126.
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 Spring '08
 staff
 Algebra, Natural number, Euclidean algorithm, ab − nq, ab mod

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