CS 101 Problem Set-2 (for practice, not to be submitted) Problems requiring iterative solutions

August 11th, 2011

Simple Problems

1. Write programs to calculate N!, for a given input value of N, using each of the three iterative control statements ’while’, ’do while’, and ’for’. 2. Given a positive integer N, write a program to find the sum of its digits. The program should also output individual digits as these are extracted. 3. We have to calculate sum of the terms of a series 1 + x + x2 + x3 ..., for a given value of x. x is postive but less than 1. Write a program to find the sum by including all those terms, such that the difference between the last two terms included in the sum is less than 1.0E-8. 4. We can calculate the nth odd number by the formula 2n - 1. Write a program to find the sum of N odd numbers for a given value of N. Compare the result with n2. 5. Write a program which accepts two numbers m and n from the user and displays the multiplication tables of m upto m × n. 6. Write a program which accepts a number n between 1 and 100 from the user and displays all numbers between 1 and 1000 not divisible by n. 7. Write a program which displays all numbers between 1 and 1000 which are either divisible by 2 or 5 but not both. 8. Write a program which will find all real roots to the following equations in the range [-10000,10000]: (a) 4x2 + 3x − 12 = 0 (b) x3 + x2 = 78 3 1 (c) 0.001x + x = −12 9. A prime number is defined as a natural number greater than 1 which is divisible by only 1 and itself. Write a program which accepts a number from the user and checks if it is a prime number. 10. We define twin primes as a pair of prime numbers which differ by 2. Write a program to display the first 25 twin primes. 11. Write a program which accepts a number n from the user and displays the smallest prime number greater than n. 12. A semiprime is defined as a natural number which can be written as a product of 2 prime numbers (not necessarily distinct). Write a program which prints the first 20 semiprimes. 13. Write a program which accepts a number n from the user and displays the largest semiprime less than n. 14. Greatest Common Divisor (gcd) of two or more non-zero integers is defined as the largest positive integer that divides the numbers without a remainder. For example, the gcd of 8 and 12 is 4.Write a program which accepts two integers from the user and finds their gcd.

1 15. Two integers are said to be coprime (or relatively prime) if they have no common positive divisor other than 1 or, equivalently, if their gcd is 1. Write a program which accepts an integer n and displays all coprime pairs (a, b) where a and b are positive integers less than n. 16. Goldbach’s Conjecture: Every even integer greater than 2 can be expressed as the sum of two primes. This fact has been verified for large values. Write a program to verify Goldbach’s conjecture upto 106.

17. Write a program which accepts a number from the user and displays the sum of digits of the number. For instance if the user enters 1789, then the output should be 25 which is 1+7+8+9. 18. Write a program which accepts a number from the user and displays the product of digits of the number. For instance if the user enters 2341, then the output should be 24 which is 2 × 3 × 4 × 1.

19. A palindromic number is a number that is the same when written forwards or backwards. For example 11, 101, 110011... Write a program which accepts a numbers and determines if it is palindromic or not. 20. Positive Integers a, b, c are pythagorean triplets if a2 + b2 = c2. Write a program which displays all pythagorean triplets (a, b, c) if c < 100.

21. If the sum of cubes of the digits of a natural number is equal to the number itself, then such a number is called an Armstrong number. For instance 153 is an Armstrong number as 153 = 13 + 53 + 33. Write a program which displays the first ten armstrong numbers. 22. A perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself. For instance the proper divisors of 6 are 1, 2 and 3. 6=1+2+3, hence 6 is a perfect number. Write a program which accepts a positive integer n and displays the number of perfect numbers less than n.

23. Fibonacci numbers are the numbers defined by the sequence an = an−1 +an−2, with a0 = 0 and a1 = 1. Write a program which accepts an integer n from the user and displays the nth fibonacci number. 24. We define a function Φ over natural numbers as follows: For every natural number n, Φ(n) is defined to be the number of positive integers less than or equal to n that are coprime to n (i.e. having no common positive factors other than 1). For instance let n = 9,then 1,2,4,5,7,8 are coprime to 9. Hence Φ(9) = 6. Write a program which accepts a number n from the user and displays Φ(n).

Challenging Problems

1. We define for every integer, n a river of integers (that is a sequence), hRmi as follows: R1 = n, and Rk = product of the digits of Rk−1, for all positive integers k > 1. Now we define the river to have stopped or terminated if Rt = 0, 1, 2..., or 9 for some positive integer t. For example let us look at the river of number 134, R1 = 134,R2 = 12,R3 = 2. Hence the river terminates at 2. The length of the river is 3. Assuming all rivers terminate for some finite t (which can be proved), write a program that accepts an integer, r from 0 to 9 from the user and determines the number of numbers less than 10000 whose river terminates at r. Also determine the length of the longest river (of a number less than 10000) which terminates at r. 2. We define a function π over natural numbers as follows: We define π(n), for a given positive integer n, as the least positive integer m such that Fm is divisible by n and Fm+1 when divided by n leaves a th th remainder 1 where Fm and Fm+1 are the m and m+1 fibonacci numbers. This is often referred to as the Pisano period. For instance let n = 4,then the first few terms of fibonacci series (1,1,2,3,5,8,13,21...) leave the following remainder when divided by 4 - 1,1,2,3,1,0,1,1. Hence π(4) = 6. Write a program which finds all solutions to the equation π(n) = π(n2) for all positive integers n less than 100. 3. Write a program which accepts 2 integers m and n and a prime p from the user and checks if there exist m primes such that the sum of these primes is equal to p − n.

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