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the remainder. When the remainder is 0, the other integer in the pair will be the greatest common divisor of the original pair (and of all the intermediate pairs). For example, if m is 532 and n is 112, then the Euclidean Algorithm reduces the pair (532,112) to (28,0) by (532,112) (112,84) (84,28) (28,0). So 28 is the greatest common divisor of 532 and 112. This result can be verified from the facts that 532 = 28 19 and 112 = 28 8. The reason that the Euclidean Algorithm works is that each pair in the sequence has the same set of divisors, which are precisely the factors of the greatest common divisor. In the example above, that common set of divisors is {1, 2, 4, 7, 14, 28}. The reason that this set of divisors is invariant under the reduction process is that when m = n q + r, a number is a common divisor of m and n if and only if it is a common divisor of n and r.
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4.1 4.2 4.3 If the control condition of a while loop is initially false, then the loop is skipped altogether; the statement(s) inside the loop are not executed at all. The control variable in a for loop has to be declared before the loop (instead of within its control mechanism) if it is used outside of the loop s statement block, as in Example 4.14 on page 67. The break statement provides better control of loops by allowing immediate termination of the loop after any statement within its block. Without a break statement, the loop can terminate only at the beginning or at the end of the block. a. The minimum number of iterations that a while loop could make is 0. b. The minimum number of iterations that a do..while loop could make is 1. 4.5 4.6 That is an infinite loop because the value of its control variable n does not change. There is no difference between the effects of those two program fragments, unless s is a break statement or s is a compound statement (i.e., a block) that contains a break statement or a continue statement. For example, this for statement will iterate 4 times and then terminate normally: for (i = 0; i < 4; i++) if (i == 2) continue; but this while statement will be an infinite loop: i = 0; while (i < 4) { if (i == 2) continue; i++; } 4.7 4.8 The constant PI is not initialized. Every constant must be initialized at its declaration. An infinite loop is one that continues without control; it can be stopped only by a branching statement within the loop (such as a break or goto statement) or by aborting the program (e.g., with Ctrl+C). Infinite loops are useful if they are stopped with branching statements. A loop can be terminated by a statement in the middle of its block by using a break or a goto statement. Floating-point variables suffer from round-off error. After undergoing arithmetic transformations, exact values may not be what would be expected. So a test such as (y == x) may not work correctly.
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4.1 4.2 First, x is initialized to 4.15 and i is initialized to 0. Then x is doubled three times by the three iterations of the for loop. The equivalent while loops are: a. while (e) s; b. while (true) { s; e; }, assuming that s contains no break or continue statements. The equivalent while loop is: int i=1; while (i <= n) { cout << i*i << " "; i++; } The output is 1 1 3 9 5 9 7 7 The output is End of program. End of program. 3 End of program. 5 End of program. End of program. a. The 23 bits hold the 2nd through 24th bit of the mantissa. The first bit must be a 1, so it is not stored. Thus 24 bits are represented. These 24 bits can hold 224 numbers. And 224 = 16,777,216, which has 7 digits with full range, so 7 complete digits can be represented. But the last digit is in doubt because of rounding. Thus, the 32-bit float type yields 6 significant digits of precision. b. The 8 bits that the 32-bit float type uses for its exponent can hold 28 = 256 different numbers. Two of these are reserved for indicating underflow and overflow, leaving 254 numbers for exponents. So an exponent can range from 126 to +127, yielding a magnitude range of 2 126 = 1.175494 10 38 to 2127 = 1.70141 1038. This program uses a while loop to compute the sum of the first n squares, where n is input: int main() { int n; cout << "Enter a positive integer: "; cin >> n; int sum=0, i=0; while (i++ < n) sum += i*i; cout << "The sum of the first " << n << " squares is " << sum << endl; } Enter a positive integer: 6 The sum of the first 6 squares is 91 This program uses a for loop to compute the sum of the first n squares, where n is input: int main() { int n; cout << "Enter a positive integer: "; cin >> n; int sum=0; for (int i=1; i <= n; i++)
4.4 4.5