If x and y are integers, what is the remainder when x^2 + y^2 is divided by 5?
1. When x-y is divided by 5, the remainder is 1
2. When x+y is divided by 5, the remainder is 2
gmat question pack 1 hard remainder
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i think the answer is E as we do not know the quotients in both the case so even if we combine both the statements we cannot get the answer.
as x-y = 5k+1
and x+y = 5r+2
so we cannot get the value of x^2 + y^2 .
but can we find the value of x and y individually thats my question and then square it before dividing by 5 ; to get the answer but can we solve the above for value of x and y .
as x-y = 5k+1
and x+y = 5r+2
so we cannot get the value of x^2 + y^2 .
but can we find the value of x and y individually thats my question and then square it before dividing by 5 ; to get the answer but can we solve the above for value of x and y .
- karthikgmat
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- karthikgmat
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assume x-y divided by 5 remainder 1
so , x-y = 5r+1
x+y divided by 5 remainder 2
x+y = 5s+2
square of (x+y) + square of (x-y) = 2(square of x+ square of y) = (5r+1)^2 + (5s+2)^2
x^2+ y^2 = [5*(r^2+s^2)+(1+4)+10r+10s] / 2
= 5*[(r^2+s^2+1+2r+2s)/2]
so it is clearly divisible y 5
so , x-y = 5r+1
x+y divided by 5 remainder 2
x+y = 5s+2
square of (x+y) + square of (x-y) = 2(square of x+ square of y) = (5r+1)^2 + (5s+2)^2
x^2+ y^2 = [5*(r^2+s^2)+(1+4)+10r+10s] / 2
= 5*[(r^2+s^2+1+2r+2s)/2]
so it is clearly divisible y 5