The arithmetic of integers
Problem 1. The positive integers from 1 to 12 have been divided into six pairs so that the sum of the two numbers in each pair is a distinct prime number. Find the largest of these prime number.
Problem 2. The sum of the squares of three prime numbers is 5070. Find the product of these three prime numbers.
Problem 3. A number is said to be strange if in its prime factorization, the power of each prime number is odd. For istance, 22,23 and 24 form a block of three consecutive strange numbers because . Find the greatest length of a block of consecutive strange numbers.
Problem 4. Find the number of consecutive 0s at the end of
Problem 5. Find the smallest positive integer which is times the square of some positive integer and also times the fifth power of some other positive integer.
Problem 6. Sum of seven consecutive positive integer is the cube of an integer and the sum of the middle three numbers is the square of an integer. Find the smallest possible value of the middle number.
Problem 7. Some factors in the product are to be removed so that the product of the remaining factors is the square of an integer. Find the minimum number of factors that must be removed. Continue reading “IMC training 2016 (1)”