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Grade: 12


For x, y, z ∈ Z, suppose that 5 divides x^2 + y^2 + z^2 . Prove that 5 divides at least one of x, y or z. Hint: What remainders can be left when the square of an integer is divided by 5?

one month ago

Answers : (1)

Aditya Gupta
2068 Points
Any number can only be of the form a= 5k+r
So a^2= 5m+r^2 so remainder can only be 0,1,4,9,16
Or 0,1,4,4,1
Or 0,1,4.
Assume to the contrary that none of x y z is divisible by 5, so r can't be 0. 
So remainder for each of x y and z can only be 1 or 4.
All possible cases for 3 such remainders are (1,1,1); (1,1,4); (1,4,4) and (4,4,4) and their sums are 3, 6, 9, 12 which are 3,1,4,2 modulo 5, meaning that x^2+y^2+z^2 can only leave a remainder of 1,2,3 or 4 when divided by 5, which is clearly a contradiction to the given fact that it is divisible by 5 so should have left a remainder of 0. Hence our initial assumption was wrong and at least one of x y z has to be divisible by 5. QED.
16 days ago
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