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main0 -t for (int i = -1; i < 8; i++ > -c for (int j = -1; j <= i +l; j++) tout << " ' -CC comb <iA>; tout CC endl; 1 1 int factorial(int); // Returns C(n,k), the number of combinations of k from n: int comb(int n, int k)

if (n c 0 II k c 0 II k > n) return 0; return factorial(n)/(factorial(k)*factorial(n-k));

Notethatthe factorial0 function must be declared above the comb ( ) function because it function because it is is used by that function. But it does not need to be declared above the main ( > not used there.

b 3 3

4.16 Write and test the dig i t ( ) function: int digit(int n, int k)

This function returns the kth digit of the positive integer n. For example, if n is the integer 29,415, then the call digit (n, 0 ) would return the digit 5, and the call dig i t ( n , 2 > would return the digit 4. Note that the digits are numbered from right to left beginning with the zeroth digit.

This removes the right-most digit of n k times. This reduces n to an integer whose right-most digit is the same as the kth digit of the original integer. That digit is then obtained as the remainder from division by 10: int digit(int, int); main0 1 int n, k; tout << "Integer: "; tin >> n; do 1 tout << "Digit: "; tin >> k; tout << "The ' << k C-C "th digit of ' -CC n << ' is ' << digit(n, k) << endl; } while (k > 0);

// Returns the kth digit of the integer n: int digit(int n, int k) for (int i = 0; i < k; i++) // remove right-most digit n /= 10; return n % 10;

c 1 4

4.17 The ancient Greeks classified numbers geometrically. For example, a number was called trian-

gular if that number of pebbles could be arranged in a symmetric triangle. The first eight triangular numbers are 1, 3,6, 10, 15,21,28, and 36:

T,= 1

d 5 3

T, = 3

T, = 6

T,= 10

T,= 15

</sequence> <sequence minOccurs="0"> <element name="Seed" type="ds:CryptoBinary"/> <element name="PgenCounter" type="ds:CryptoBinary"/> </sequence> </sequence> </complexType> DTD: <!ELEMENT <!ELEMENT <!ELEMENT <!ELEMENT <!ELEMENT <!ELEMENT <!ELEMENT <!ELEMENT DSAKeyValue (P, Q, G, Y, J , (Seed, PgenCounter) ) > P (#PCDATA) > Q (#PCDATA) > G (#PCDATA) > Y (#PCDATA) > J (#PCDATA) > Seed (#PCDATA) > PgenCounter (#PCDATA) >

e 2 4

Write and test the boolean function:

int isTriangular(int n)

The argument n is triangular if and only if it is a sum of consecutive integers 1 + 2 + 3 + . So we just have to compute these sums until we find one that is greater than or equal to n. If that sum is equal to n, then n is a triangular number; otherwise, it isn t:

int isTriangular(int); main0 int n; do -t tin >> n; if (isTriangular( tout -CC n << ' is triangular.\n"; else tout CC n << ' is not triangular.\n"; } while (n > 0);

f 5 4

// Returns 1 i f n i s a triangular int isTriangular(int n) 1 int i = 0, sum = 0; while (sum < n) sum += ++i ; if (sum == n) return 1; else return 0;

number (1,

10, 15, etc.):

We assume that the max ( int , int max(int, int); in t ) function is already available:

g 6 2

int max(int x, int y, int z) -t int max(int,int); return max(max(x,y),z); )

Every point in the coordinate plane has a unique pair (x, y) of rectangular coordinates and a unique pair (r, 0) of polar coordinates with r 2 0 and 0 5 0 < 27~. The following function converts from rectangular to polar coordinates. Since the output consists of more than one variable, the two output variables r and t are passed by reference:

void rectangularToPolar(double&

r, double& t, double x, double y)

h 1 6