IMC training 2016 (1)

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 $22={{2}^{1}}\times {{11}^{1}},23={{23}^{1}},24={{2}^{3}}\times {{3}^{1}}$. Find the greatest length of a block of consecutive strange numbers.

Problem 4. Find the number of consecutive 0s at the end of $2003!=1\times 2\times 3\times ...\times 2003.$

Problem 5. Find the smallest positive integer which is $2$ times the square of some positive integer and also $5$ 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 $1\times 2\times 3\times 4\times ...\times 27$ 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.

Problem 8. Do there exist $2016$integers such that their sum and their product are both equal to $2016$? If so, give an example. If not, give an explanation.

Problem 9. Find the least positive integer $n$ such that no matter how ${{10}^{n}}$is expressed as a product of two positive integers, at least one of these two integers contains the digit $0.$

Problem 10. Ampang Street has no more than $15$ houses, numbered $1,2,3$ and so on. Mrs. Lau lives in one of the houses, but not the first or the last. The product of the numbers of all the houses before hers is the same as the product of the numbers of all the houses after hers. Find the number of houses on the Ampang Street.

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