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gmatclub last digit of integer x^2 - y^2 a zero

by arora007 » Tue Jul 06, 2010 8:33 am
Is the last digit of integer x^2 - y^2 a zero?

1. x - y is an integer divisible by 30
2. x + y is an integer divisible by 70

OA is C, how easily we assume sometimes in algebra everything mentioned is an integer, when its not
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by albatross86 » Tue Jul 06, 2010 8:45 am
IS the last digit of INTEGER (x^2 - y^2) a zero?

1. x- y is an integer divisible by 30. Lets call it 30n where n is a positive integer.

Thus x^2 - y^2 = (x + y)*(x - y) = (x + y) * 30n

Now if (x + y) is an integer, it will lead to (x^2 - y^2) having units digit of zero.

eg. x = 60, y = 30; x - y = 30; x + y = 90 => x^2 - y^2 = 2700

However if (x+y) is non-integer, this would lead to (x^2 - y^2) possibly having a non-zero units digit.

eg. x = 60.3, y = 30.3 => x - y = 30 but x + y = 90.3 => x^2 - y^2 = 2709

INSUFFICIENT

2. x + y is an integer divisible by 70. Let's call it 70m.

This is exactly analogous to statement 1, and we will still be able to choose non-integer values of x and y that lead to non-zero units digits in x^2 - y^2

INSUFFICIENT

Both 1 and 2:

x - y = 30n and x + y = 70m

Thus x^2 - y^2 = 2100mn

m and n are positive integers, so we can be sure that x^2 - y^2 has a units digit of zero.

SUFFICIENT

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