If x and y are distinct positive integers and x + y is even, what is t

This is a question based on the divisibility rule for 10, which depends on the units digit of the dividend. Since there is a power involved in the expression, you also need to know the unit digit cycles to be able to solve this question.

It is given that x+y is even. Therefore, \((x+y)^a\) will also be even, regardless of the value of a. This also means that the unit digit of \((x+y)^a\) can be either 2 or 4 or 6 or 8 or 0, depending on the unit digit of (x+y) and the value of a.
Therefore, any data that gives us concrete information about all these 3 variables will be sufficient data.

From statement I, we get the value of y to be 6, but we neither have the value of x nor a. So, how does this affect the answer?

If y = 16 and x = 14, (x+y) = 30; the power 'a' does not matter in this case since 30 raised to the power of any number will give us the unit digit as 0. As such, the remainder when \((x+y)^a\) is divided by 10, will be 0.

But, if y = 16 and x = 12, (x+y) = 28. In this case, the unit digit of \((x+y)^a\) depends on the value of a, which is unknown.

Hence, statement I is insufficient. Answer options A and D can be eliminated. Possible answer options are B, C or E.

From statement II alone, \((xy)^a\) is divisible by 10. This means that the product xy is a number ending with 0. However, we do not know what the numbers x and y themselves are.

x = 6 and y = 10 is a possible set of values. For this set, \((x+y)^a\) will leave a remainder when divided by 10.

x = 10 and y = 20 is another possible set. For this set, \((x+y)^a\) will not leave any remainder when divided by 10.

Statement II is insufficient. Answer option B can be eliminated. Possible answer options are C or E.

From statements I and II together, we know that unit digit of y is 6 and also product of x and y ends with a 0. Therefore, unit digit of x should be 0, which in turn means that the unit digit of (x+y) will be 6.

When the units digit of a number is 6, it will give the units digit as 6 whatever power we may raise it to. Therefore, units digit of \((x+y)^a\) = 6. When this number is divided by 10, the remainder will be 6, since, for divisibility by 10, we check the units digit of the number.

A point to note here is that, when analyzing the second statement alone, it’s very easy to fall into the trap of assuming that x and y are single digit numbers. That’s not mentioned anywhere in the question, so assuming that would be false.

Hope this helps!