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Even Numbers Created by Inna Shapiro ©2008 Problem 1 The sum of two integers is even. What is true about the product of those two integers? Is it even or odd? Answer The sum of two integers is even, if (1) both integers are even, or (2)both integers are odd. That means the product could be ether even (1), or odd (2). Problem 2 The sum of three integers is even. What is true about the product of those integers? Is it even or odd? Answer The sum of three integers is even, if (1) all three integers are even, or (2)two integers are odd and the third one is even. The product is even in both cases. Problem 3 The integer A can be written only with digit 4, for example 444,444. The integer B can be written only with digit 3, for example 33,333. Is it possible that A is divisible by B? Is it possible that B is divisible by A? Answer 1) A can be divisible by B, for example, 444,444/33 = 13,468 2) B cannot be divisible by A, because B is always an odd number, and so it cannot be divisible by even number. Problem 4 A teacher wrote number 20 on a blackboard and suggested that each student in turn erase the number on the board and replace it with a number bigger than it by 1 or smaller than it by 1. There are 33 students in a room. Can the final number be equal to 10, if each student does it once? Answer No, the result cannot be 10. It must be an odd number. 20 is even number. The first student will make it odd, the second – even, and so on. The 33rd student will write an odd number. Problem 5 A teacher wrote four numbers on a blackboard: 0, 1, 0, 0. A student can add 1 to any two of the numbers, as many times as he wants. Can the student get all four numbers to be equal? Answer The sum of the original four numbers is odd. Adding 1 to any two numbers leaves the sum odd. That means the student cannot get four equal numbers, because then their sum would be even. Problem 6 There are 25 gallons of milk in a bucket. A farmer has empty bottles of different volume – 1 gal, 3 gal and 5 gal. Can he fill ten bottles with that milk? Answer No, he can’t. 1, 3 and 5 are all odd numbers. Any two bottles together would contain an even quantity of gallons. That means that any ten bottles also have to contain an even quantity gallons of milk. Problem 7 Max said that he knows four integers such that the sum and the product of these integers are odd. Jenny said that his calculations must be wrong. Who is right? Answer Jenny is right. If any of the four integers is even, the product would be even. If the product is odd, that means all four integers are odd. Therefore the sum of those four odd integers is even. Problem 8 There are 11 plates on a round table. Mary tried to place cherries in these plates so that the numbers of cherries on any two adjacent plates differ by one. Can she do it? Answer No, she can’t. Suppose there is an odd number of cherries on the first plate. Then the second plate contains an even number of cherries. And so on. The eleventh plate contains an odd number of cherries, and the adjacent first plate also contains an odd number, so these numbers cannot differ by 1. Same happens if there is an even number of cherries on the first plate. Problem 9 The are 17 numbers on a screen – 1, 2, 3, …17. Can you place “+” and “-” signs between those numbers to get the value of the expression to be 0? Answer No, you can’t do that. The first number 1 is odd. After adding or subtracting an even number 2, the result is still odd. After adding or subtracting an odd number 3, the result changes to an even number. And so on. There are nine odd numbers in the row: 1, 3, 5, 7, 9, 11, 13, 15, 17. After adding or subtracting the last number 17 the result must be odd and consequently cannot be equal to 0. Problem 10 The are 2005 numbers on a screen – 1 , 2, 3, … 2004, 2005. Can you place “+” and “-” signs between those numbers to get 0 as a result of calculation? Answer No, you can’t do that. The first number 1 is odd. After adding or subtracting the even number 2, the result is still odd. After adding or subtracting the odd number 3, the result changes to even number. And so on. There are 1003 odd numbers on the screen. After adding or subtracting the last number, 2005, the result must be odd, and so it cannot be equal to 0.