If x and y are different positive integers, is x − y divisible by 10?

statement 1:
unit digits of x^2 and y^2 are same.
unit digit of a square can be [0,1,4,5,6,9]
x and y can have different unit digits.
x = 2: x^2 = 4
y = 8: y^2 = 64
not sufficient

statement 2:
unit digit of x = 1, but we do not know anything about y.
not sufficient

combining both statements,
x = 1: x^2 = 1
y^2 unit digit is 1: y can be [1,9]
when (x,y) is (1,1): pair is divisible by 10
when (x,y) is (1,9): pair is not divisible by 10
not sufficient
Ans: E

Explanation:
x-y divisible by 10?
Statement 1: Unit digits of x^2 is the same as the units digit of y^2.
Let take example x=11 , 21.. y =1 ; x=17, y=13 or any example.
x-y may be or may not divisible by 10

Not Sufficient

Statement 2: The units digit of x is 1
So, x can be 11, 21,31... we don't know any thing about y.

Not Sufficient.

From 1 & 2,
x unit digit will be 1, also x^2 & Y^2 will be 1 & x and y are different positive integers.
It will be divisible by 10 always
IMO-C

IMO E

(x-y) will be divisible by 10 is last digit of (x-y) is zero.

(1) The units digit of x^2 is the same as the units digit of y^2.

Let x = 8^2 & y=2^2 (same last digit as 4)
but (x-y)= 6 is not divisible by 10

But, if x=y, than (x-y) is divisible by 10

This statement is not sufficient.

(2) The units digit of x is 1.

Since no information about y,
This statement is not sufficient.

Together:
Let x = 11 & y = 9 (Unit digit of x is 1 & x^2 = 121 & y^2=81 both have same last digit)
But (x-y) = 2 not divisible by 10.
But if, x=11 & y=11 , (x-y) is divisible by 10

Not Sufficient.

S1: We know x and y are both integers and will thus both be perfect squares, and can thus use this to create test values.

Case 1: x^2 = 36, y^2 = 16, so x = 6, y = 4. 6-4 = 2, which is NOT divisible by 10.

Case 2: x^2 = 225, y^2 = 25, so x = 15, y = 5. 15 - 5 = 10, which IS divisible by 10

NOT SUFFICIENT.
__

S2: We don't know the value of y.

NOT SUFFICIENT.
__

S1+S2: So now we know that x^2 and y^2 are both perfect squares ending with 1.

Case 1: x^2 = 121, y^2 = 81, so x = 11, y = 9. 11 - 9 = 2, which is NOT divisible by 10.

Case 2: x^2 = 121, y^2 = 1, so x =11, y = 1. 11 - 1 = 10, which IS divisible by 10.

NOT SUFFICIENT
__

ANSWER: E

Statement:1 The units digit of x^2 is the same as the units digit of y^2.
So x^2 can be 36 and y^2 can be 16, so x-y will be 6-4 that is 2,
Also x^2 can be 196 and y^2 can be 16, so x-y will be 14-4 that is 10, hence true . Therefore Statement 1 is insufficient.
Statement 2: The units digit of x is 1.
No information about y. Hence x-y could be anything, therefore Statement 2 is insufficient.

Now Combining both the statements:
So, by that the possible value of x can be "1","11","21","31" ...and so on.
And by statement 2, the square of "X" will always end with "1"
the possible value of y^2 can be "81","121","441"..."41^2","15^2".....
Now for all the values of "Y" the statement will hold true except "Y^2 = 81 and Y = 9" as then x-y will not be divisible by 10 say (11-9). Hence both of them are also insufficient together as well.
Hence "E" should be the answer

Ans - C

Statement- 1
Multiple sets can be possible like
X,Y = 5,5. 6,6 , 11,1, Where x-y is divisible
Also 6,5 where x-y is not, so insufficient

Statement 2
Possiblex - 1,11,21,31...... all will yield unit digit 1, but still y value will determine, so this is insufficient

With 1-2
If x has 1 as unit digit, y will also have 1 as unit, this will always yield x-y = abc0, 0 in unit place, so our x-y is divisible by 10

Sufficient

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Q.
If x and y are different positive integers, is x − y divisible by 10?

(1)
If x=11 and y=1, (x − y) is divisible by 10.
If x=11 and y=9, (x − y) is NOT divisible by 10.
NOT SUFFICIENT

(2)
We don't know about y.
NOT SUFFICIENT

(1)+(2)
If x=11 and y=1, (x − y) is divisible by 10.
If x=11 and y=9, (x − y) is NOT divisible by 10.
NOT SUFFICIENT

FINAL ANSWER IS (E)

If x and y are different positive integers, is x − y divisible by 10?

(1) The units digit of x^2 is the same as the units digit of y^2.
(2) The units digit of x is 1.

1) x^2 units digit may be 9, x can be 3 or 7
y^2 units digit is also same as x, so y can also be 3 or 7
insufficient

2) units digit of x is 1
nothing about y
insufficient

1+2)
x^2 units digit is 1, y^2 units digit is also 1,
so y can have units digit as 9 or 1
insufficient

Ans E

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