Solutions manual for NTB - 3.3. Basic integer arithmetic
Let's review some notations and facts. For an integer 
, we define its bit length, or simply, its length, which we denote by 
, to be cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 number of bits in cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 binary representation of 
; more precisely,
, we say that is an 
-bit integer. Notice that if is a positive, -bit integer, cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365n 
, or equivalently, 
.
Let a and b be arbitrary integers. Then we have:
(i) We can compute
in time 
.
(ii) We can compute
in time 
.
Let a and b be arbitrary integers. Then we have:
(i) We can compute in time .
(ii) We can compute in time .
(iii) If
, we can compute cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 quotient 
and cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 remainder 
in time 
.
Exercise 3.24. Let
with 
, and let 
. Show that:
Proof
If you look at cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 list of errata, you'll see that I found a stronger version of this exercise. My proof is as following.
We see that
. Hence what we need to prove is:
which is cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 same as proving:
for some 
which in turn can be proved by using this fact:
for all .
(q.e.d.)
Exercise 3.25. Let
be postivie integers. Show that:
Proof
As in Exercise 3.24, I think I found a slightly better version of this exercise as follows:
We can prove it using cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 same technique as cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 proof of Exercise 3.24. We see that:
,
,
Since we have for all , we see that:
and,
.
(q.e.d.)
Exercise 3.26. Show that given integers, with each 
, we can compute cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 product 
in time 
.
Proof
(q.e.d.)
Exercise 3.27. Show that given integers
, with each 
, where 
, we can compute 
in time .
Proof
(q.e.d.)
Exercise 3.28. Show that given integers, with each , we can compute 
, where , in time .
Proof
(q.e.d.)
Exercise 3.29. This exercise develops an algorithm to compute
for a given positive integer 
. Consider cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 following algorithm:
(a) Show that this algorithm correctly computes.
(b) In a straightforward implementation of this algorithm, each loop iteration takes time, yielding a total running time of 
. Give a more careful implementation, so that each loop iteration takes time 
, yielding a total running time is .
Proof
(q.e.d.)
Exercise 3.30. Modify cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 algorithm in cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 previous exercise so that given positive integers and 
, with 
, it computes 
in time 
.
Proof
(q.e.d.)
Exercise 3.31. An integer
is called a perfect power if 
for some integers 
and 
. Using cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 algorithm from cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 previous exercis, design an efficient algorithm that determines if a given is perfect power, and if it is, also computes and 
such that , where , , and is as small as possible. Your algorithm should run in time 
where 
.
Proof
(q.e.d.)
Exercise 3.32. Show how to convert (in both directions) in time between cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 base-10 representation and our implementation's internal representation of an integer .
Proof
(q.e.d.)
, we define its bit length, or simply, its length, which we denote by , to be cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 number of bits in cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 binary representation of ; more precisely, , we say that is an -bit integer. Notice that if is a positive, -bit integer, cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365n , or equivalently, .Let a and b be arbitrary integers. Then we have:
(i) We can compute
in time .(ii) We can compute
in time .Let a and b be arbitrary integers. Then we have:
(i) We can compute
in time .(ii) We can compute
in time .(iii) If
, we can compute cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 quotient and cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 remainder in time .Exercise 3.24. Let
with , and let . Show that:Proof
If you look at cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 list of errata, you'll see that I found a stronger version of this exercise. My proof is as following.
We see that
. Hence what we need to prove is:which is cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 same as proving:
for some which in turn can be proved by using this fact:
for all .(q.e.d.)
Exercise 3.25. Let
be postivie integers. Show that:Proof
As in Exercise 3.24, I think I found a slightly better version of this exercise as follows:
We can prove it using cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 same technique as cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 proof of Exercise 3.24. We see that:
, ,Since we have
for all , we see that:and,
.(q.e.d.)
Exercise 3.26. Show that given integers
, with each , we can compute cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 product in time .Proof
(q.e.d.)
Exercise 3.27. Show that given integers
, with each , where , we can compute in time .Proof
(q.e.d.)
Exercise 3.28. Show that given integers
, with each , we can compute , where , in time .Proof
(q.e.d.)
Exercise 3.29. This exercise develops an algorithm to compute
for a given positive integer . Consider cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 following algorithm:(a) Show that this algorithm correctly computes
.(b) In a straightforward implementation of this algorithm, each loop iteration takes time
, yielding a total running time of . Give a more careful implementation, so that each loop iteration takes time , yielding a total running time is .Proof
(q.e.d.)
Exercise 3.30. Modify cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 algorithm in cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 previous exercise so that given positive integers
and , with , it computes in time .Proof
(q.e.d.)
Exercise 3.31. An integer
is called a perfect power if for some integers and . Using cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 algorithm from cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 previous exercis, design an efficient algorithm that determines if a given is perfect power, and if it is, also computes and such that , where , , and is as small as possible. Your algorithm should run in time where .Proof
(q.e.d.)
Exercise 3.32. Show how to convert (in both directions) in time
between cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 base-10 representation and our implementation's internal representation of an integer .Proof
(q.e.d.)