Let
n
=
p
1
. . . p
r
=
q
1
. . . q
s
, where none of the
p
i
equal any of the
q
j
and
p
1
is the smallest
among all the primes. Then using the division algorithm to divide
q
1
by
p
1
, we can find a
smaller number
k
which also has two prime factorizations, which is a contradiction. This
works because we know that the remainder in this division is less than
p
1
; in number systems
where remainders don’t work this way we don’t have unique factorization.
8.
We have seen that an infinite decimal corresponds to a rational number if and only if
it is eventually repeating. If instead we consider real numbers expressed in some other base,
is this still true? Explain why or why not.
Solution.
On the one hand, if we have an infinite “decimal” in another base, we can sum
a geometric series to find the rational number it represents. On the other hand, if we have
two integers and we divide them to find the “decimal” expansion, there are only a finite
number of remainders which are possible, so the remainders must eventually repeat, and the
expansion is eventually repeating. The answer is yes.
9.
Recall the RSA algorithm for cryptography, here reproduced as it appeared on Home
work 3:
1. Choose two distinct primes
p
and
q
. (These are usually of similar size, and large; in
our examples they will be small.)
2. Compute
n
=
pq
; this will be the modulus.
3. Compute
φ
(
n
) = (
p

1)(
q

1), the
totient
of
n
. This is the number of numbers among
1
,
2
, . . . , n

1 which have no factor in common with
n
.
4. Choose an exponent
e
such that 1
< e < φ
(
n
), where
e
and
φ
(
n
) are relatively prime.
e
is the
encoding exponent
.
5.
Find
d
such that
de
≡
1
(mod
φ
(
n
)), by either trial and error or the Euclidean
algorithm.
d
is called the
decoding exponent
.
Then the
public key
(which is used to encode messages to be sent to you) is the pair of
numbers (
n, e
); the
private key
is the single number
d
. To encrypt the message
W
, compute
C
=
W
e
(mod
n
), and send that.
To decode the ciphertext
C
, compute
C
d
(mod
n
).
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 Summer '09
 Lugo
 Math, Solution., Bank identification number, prime factorizations

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