🚧 In mid of 4.10
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2 changed files with 161 additions and 2 deletions
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@ -9175,10 +9175,14 @@ executed.
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$i := 2\\ \text{\textbf{if }} (i > 3 \text{ or } i \leq 0)\\ \ \ \ \ \text{\textbf{then }} z := 1\\ \ \ \ \ \text{\textbf{else }} z := 0$
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$z = 0$
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2.
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$i := 3\\ \text{\textbf{if }} (i \leq 3 \text{ or } i > 6)\\ \ \ \ \ \text{\textbf{then }} z := 2\\ \ \ \ \ \text{\textbf{else }} z := 0$
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$z = 2$
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3. Consider the following algorithm segment:
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$\text{\textbf{if }} x \cdot y > 0 \text{\textbf{ then do }} y := 3 \cdot x\\ \ \ \ \ x := x + 1 \text{\textbf{end do}}\\ \ \ \ \ z := x \cdot y$
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@ -9188,8 +9192,12 @@ below.
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a. $x = 2, y = 3$
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$y = 3 \cdot 3 = 9$, and $x = 2 + 1 = 3$, and $z = 9 \cdot 3 \cdot = 27$
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b. $x = 1, y = 1$
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$y = 3 \cdot 1 = 3$, and $x = 1 + 1 = 2$, and $z = 3 \cdot 2 = 6$
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Find the values of $a$ and $e$ after execution of the loops in 4 and 5 by first
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making trace tables for them.
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@ -9197,17 +9205,50 @@ making trace tables for them.
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$a := 2\\ \text{\textbf{for }} i := 1 \text{\textbf{ to }} 3\\ \ \ \ \ a:= 3a + 1\\ \text{\textbf{next }} i$
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| | 0 | 1 | 2 | 3 |
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| --- | - | - | -- | -- |
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| $a$ | 2 | 7 | 22 | 67 |
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| $i$ | 1 | 2 | 3 | 4 |
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After execution, $a = 67$.
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5.
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$e := 2, f := 0\\ \text{\textbf{for }} k := 1 \text{\textbf{ to }} 3\\ \ \ \ \ e := e \cdot k\\ \ \ \ \ f := e + f\\ \text{\textbf{next }} k$
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| | 0 | 1 | 2 | 3 |
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| --- | - | - | - | -- |
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| $e$ | 2 | 2 | 4 | 12 |
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| $f$ | 0 | 2 | 6 | 18 |
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| $k$ | 1 | 2 | 3 | 4 |
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After execution, $e = 12$, $f = 18$.
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Make a trace table to trace the action of Algorithm 4.10.1 for the input
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variables given in 6 and 7.
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6. $a = 26, d = 7$
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| | 0 | 1 | 2 | 3 |
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| --- | -- | -- | -- | -- |
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| $a$ | 26 | 26 | 26 | 26 |
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| $d$ | 7 | 7 | 7 | 7 |
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| $r$ | 26 | 19 | 12 | 5 |
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| $q$ | 0 | 1 | 2 | 3 |
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After execution, $q = 3$, and $r = 5$.
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7. $a = 59, d = 13$
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| | 0 | 1 | 2 | 3 | 4 |
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| --- | -- | -- | -- | -- | -- |
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| $a$ | 59 | 59 | 59 | 59 | 59 |
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| $d$ | 13 | 13 | 13 | 13 | 13 |
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| $r$ | 59 | 46 | 33 | 20 | 7 |
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| $q$ | 0 | 1 | 2 | 3 | 4 |
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After execution, $q = 4$, $r = 7$.
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8. The following algorithm segment makes change; given an amount of money $A$
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between 1¢ and 99¢, it determines a breakdown of $A$ into quarters $(q)$,
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dimes $(d)$, nickels $(n)$, and pennies $(p)$.
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@ -9223,39 +9264,104 @@ $$
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a. Trace this algorithm segment for $A = 69$.
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| | | | | |
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| --- | -- | -- | - | - |
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| $A$ | 69 | 19 | 9 | |
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| $q$ | 2 | | | |
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| $d$ | | 1 | | |
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| $n$ | | | 1 | |
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| $p$ | | | | 4 |
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b. Trace this algorithm segment for $A = 87$.
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| | | | | |
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| --- | -- | -- | - | - |
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| $A$ | 87 | 12 | 2 | |
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| $q$ | 3 | | | |
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| $d$ | | 1 | | |
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| $n$ | | | 0 | |
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| $p$ | | | | 0 |
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Find the greatest common divisor of each of the pairs of integers in 9-12. (Use
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any method you wish.)
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9. $27$ and $72$
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$$ \text{gcd}(27, 72) = 9 $$
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10. $5$ and $9$
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$$ \text{gcd}(5, 9) = 1 $$
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11. $7$ and $21$
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$$ \text{gcd}(7, 21) = 7 $$
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12. $48$ and $54$
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$$ \text{gcd}(54, 48) = \text{gcd}(48, 6) = \text{gcd}(6, 0) = 6 $$
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Use the Euclidean algorithm to hand-calculate the greatest common divisors of
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each of the pairs of itnegers in 13-16.
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13. $1,188$ and $385$
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$$ \text{gcd}(1188, 385) = \text{gcd}(385, 33) = \text{gcd}(33, 22) = \text{gcd}(22, 11) = \text{gcd}(11, 0) = 11 $$
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14. $509$ and $1,177$
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$$ \text{gcd}(1177, 509) = \text{gcd}(509, 159) = \text{gcd}(159, 32) = \text{gcd}(32, 31) = \text{gcd}(31, 1) = \text{gcd}(1, 0) = 1 $$
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15. $832$ and $10,933$
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$$ \text{gcd}(10933, 832) = \text{gcd}(832, 117) = \text{gcd}(117, 13) = \text{gcd}(13, 0) = 13 $$
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16. $4,131$ and $2,431$
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$$ \text{gcd}(4131, 2431) = \text{gcd}(2431, 1700) = \text{gcd}(1700, 731) = \text{gcd}(731, 238) = \text{gcd}(238, 17) = \text{gcd}(17, 0) = 17 $$
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Make a trace table to trace the action of Algorithm 4.10.2 for the input
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variables given in 17-19.
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17. $1,001$ and $871$
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| | | | | | | | |
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| ------------ | ---- | --- | --- | -- | -- | -- | -- |
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| $A$ | 1001 | | | | | | |
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| $B$ | 871 | | | | | | |
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| $a$ | 1001 | 871 | 130 | 91 | 39 | 13 | |
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| $b$ | 871 | 130 | 91 | 39 | 13 | 0 | |
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| $r$ | 871 | 130 | 91 | 39 | 13 | 0 | |
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| $\text{gcd}$ | | | | | | | 13 |
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After execution, $\text{gcd} = 13$.
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18. $5,859$ and $1,232$
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| | | | | | | | | |
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| ------------ | ---- | ---- | --- | --- | -- | -- | - | - |
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| $A$ | 5859 | | | | | | | |
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| $B$ | 1232 | | | | | | | |
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| $a$ | 5859 | 1232 | 931 | 301 | 28 | 21 | 7 | |
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| $b$ | 1232 | 931 | 301 | 28 | 21 | 7 | 0 | |
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| $r$ | 1232 | 931 | 301 | 28 | 21 | 7 | 0 | |
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| $\text{gcd}$ | | | | | | | | 7 |
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After execution, $\text{gcd} = 7$.
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19. $1,570$ and $488$
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| | | | | | | | | | |
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| ------------ | ---- | --- | --- | -- | -- | -- | -- | - | - |
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| $A$ | 1570 | | | | | | | | |
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| $B$ | 488 | | | | | | | | |
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| $a$ | 1570 | 488 | 106 | 64 | 42 | 22 | 20 | 2 | |
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| $b$ | 488 | 106 | 64 | 42 | 22 | 20 | 2 | 0 | |
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| $r$ | 488 | 106 | 64 | 42 | 22 | 20 | 2 | 0 | |
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| $\text{gcd}$ | | | | | | | | | 2 |
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After execution, $\text{gcd} = 2$.
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**Definition: Integers $a$ and $b$ are said to be **relatively prime** if, and
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only if, their greatest common divisor is $1$.
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@ -9264,8 +9370,34 @@ integers are relatively prime.
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20. $4,167$ and $2,563$
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| | | | | | | | | | | |
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| ------------ | ---- | ---- | ---- | --- | --- | --- | -- | - | - | - |
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| $A$ | 4167 | | | | | | | | | |
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| $B$ | 2563 | | | | | | | | | |
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| $a$ | 4167 | 2563 | 1604 | 959 | 645 | 314 | 17 | 8 | 1 | |
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| $b$ | 2563 | 1604 | 959 | 645 | 314 | 17 | 8 | 1 | 0 | |
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| $r$ | 2563 | 1604 | 959 | 645 | 314 | 17 | 8 | 1 | 0 | |
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| $\text{gcd}$ | | | | | | | | | | 1 |
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After execution, $\text{gcd} = 1$, and because of this, $4,167$ and $2,563$ are
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_relatively_ prime.
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21. $34,391$ and $6,728$
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| | | | | | | | | | |
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| ------------ | ----- | ---- | --- | --- | -- | - | - | - | - |
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| $A$ | 34391 | | | | | | | | |
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| $B$ | 6728 | | | | | | | | |
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| $a$ | 34391 | 6728 | 751 | 720 | 31 | 7 | 3 | 1 | |
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| $b$ | 6728 | 751 | 720 | 31 | 7 | 3 | 1 | 0 | |
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| $r$ | 6728 | 751 | 720 | 31 | 7 | 3 | 1 | 0 | |
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| $\text{gcd}$ | | | | | | | | | 1 |
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After execution, $\text{gcd} = 1$, and because of this, $34,391$ and $6,728$ are
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_relatively_ prime.
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RESUME HERE
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22. Prove that for all positive integers $a$ and $b$, $a \mid b$ if, and only
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if, $\text{gcd}(a, b) = a$. (Note that to prove "$A$ if, and only if, $B$,"
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you need to prove "if $A$ then $B$" and "if $B$ then $A$.")
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