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PrintEstonian Mathematical Olympiad
Estonia number theory
Problem
Non-negative integers , , , are all less than and satisfy conditions and . Dividing by gives a remainder of . Can we be certain that dividing by gives a remainder of ?
Solution
Solution 1: The numbers and satisfy the conditions of the problem, but dividing by gives a remainder of , not .
Solution 2: The numbers , and satisfy the conditions of the problem, but dividing by gives a remainder of , not .
Solution 3: The numbers , , and satisfy the conditions of the problem, but dividing by gives a remainder of , not .
Solution 4: The numbers , and satisfy the conditions of the problem, but dividing by gives a remainder of , not .
Solution 2: The numbers , and satisfy the conditions of the problem, but dividing by gives a remainder of , not .
Solution 3: The numbers , , and satisfy the conditions of the problem, but dividing by gives a remainder of , not .
Solution 4: The numbers , and satisfy the conditions of the problem, but dividing by gives a remainder of , not .
Final answer
No
Techniques
Modular ArithmeticIntegers