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The question stem DOES NOT tell us that Y is either a Single Digit No. or a two digit number..
(1) The unit digit of y^2 = 1 ... Y therefore could be any one of the following numbers:
1, 9, 11 (11 x 11 = 121 , unit digit of 121 is one), 19, 21, 29....
So statement One is not suff.
(2) The units digit of Y does not equal 1.
Taking 1 and 2 together we can remove the following numbers from the list :
1,11,21,31 etc ..
This still leaves, 9 , 19, 29, 39..The squares of all result in a NUMBER whose UNIT digit is 1 ..
Why is the official answer ASSUMING that Y has only one digit when it is not mentioned?
Read the question carefully. They are asking the units digit of Y. Not the number Y itself. 9, 19, 29, 39,..... are all values of Y; agreed. But the units digit will always remain 9. Thus, answer is C.
Agree with TH03 response, we need to concentrate on the unit digit as the questions ask for its value in the question.
(1) The unit digit of y^2 = 1 .
This comes down to either to 1 or 9 as unit digit (the number could be 1,11,21 or 9,19,29 etc) Not sufficient (2) The units digit of Y does not equal 1. This could be anything from 0, 9 except 1 Not suff.
Combining both of them we reach the answer is (C) as 9 is the last digit.