Find a solution of the equation \(x^2 + 2017 = y^2\) in integer numbers.
Is it possible to arrange some group of distinct numbers in a circle so that each number equals the sum of its neighbours?
Sarah is writing down natural numbers starting from 2. She notices that each time she writes the next number the sum of all written numbers is less than their product. She believes she can find such 57 natural numbers (not necessarily different from each other) that their sum will be greater than their product. Do you think it is possible?
(a) Can you find a set of distinct numbers which can be arranged in a circle in such a way that each number equals the product of its neighbours?
(b) Is it true that each solution of Example 1 is determined by the values of two neighbouring numbers?
It’s not that difficult to find a set of \(57\) integers which has a product strictly larger or strictly smaller than their sum. Is it possible to find \(57\) integers (not necessarily distinct) with their sum being equal to their product?
Bella spent 10 minutes searching for a 3-digit number such that it has the product of it’s digits equal to 26. She examined all 3-digit numbers one by one. Do you think she missed a possible example or is it the case that there are simply no such 3-digit numbers?
Assume you have a chance to play the following game. You need to put numbers in all cells of a \(10\times10\) table so that the sum of numbers in each column is positive and the sum of numbers in each row is negative. Once you put your numbers you cannot change them. You need to pay £1 if you want to play the game and the prize for completing the task is £100. Is it possible to win?
After proving there are no 3-digit numbers with the product of digits equal to 26 (see Example 1) Bella decided to find a 4-digit number with the product of digits equal to 98. Can she succeed in finding such a number?
Bella was encouraged by the fact that she fully understood the general concept about the existence of a number with given value of product of digits. Therefore, she started thinking about the following problems:
(a) Is there a 3-digit number with the sum of digits equal to 24?
(b) Is there a 4-digit number with the sum of digits equal to 37?
Solve these questions.
Once again consider the game from Example 2.
(a) Will you change your answer if the field is a rectangle?
(b) The rules are changed. Now you win if the sum of numbers in each row is greater than 100 and the sum of the numbers in each column is less than 100. Is it possible to win?