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We know that 2x+8y is divisible by 2 for integers x=a and y=b, so we can pick non-integer x and y so that 2x+8y=31. Since this isn't an integer solution, we get a contradiction. Therefore, no integer solution exists.
Since 2x+8y is divisible by 2 for integer x=a and y=b, but 2x+8y=31 which isn't divisible by 2, we get a contradiction. So 2x+8y=31 has no integer solutions
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