Re: If a and b are positive integers with different units digits, and b...
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16 Jun 2018, 05:22
Official Solution:-
Step 1: Analyze the Question Stem -
If a and b are positive integers with different units digits, and b is the square of an integer, is a also the square of an integer?
This is a Yes/No question. The given information provides a clue about the two numbers having different units digits and
tells you that b is a perfect square. On advanced number properties questions, it’s worth checking if two different types of
clues work together. Units digits of certain groups of numbers often fall into consistent patterns, and squares certainly fit into
that category. The units digits of the squares of 1–10 are {1, 4, 9, 6, 5, 6, 9, 4, 1, 0}, and this pattern will continue, since the
units digits of 11–20, 21–30, and so on will be the same as those of 1–10. So a perfect square can only have one of six units
digits: 0, 1, 4, 5, 6, or 9. For a to be a perfect square, its units digit must be one of those numbers.
Step 2: Evaluate the Statements
Evaluating Statement (1), you see that the sum of the integers a and b has a units digit of 8. Work your way through the list of
possible units digits of squares of integers. Again, that list is 0, 1, 4, 5, 6, and 9. If the units digit of b is 0, then the units digit
of a is 8, and 8 is not on the list. If the units digit of b is 1, the units digit of a is 7, and 7 is also not on the list. If the units
digit of b is 4, the units digit of a is 4, but this is not permitted because the units digits of a and b must be different. If the
units digit of b is 5, the units digit of a is 3, and 3 is not on the list. If the units digit of b is 6, the units digit of a is 2, and 2 is
not on the list. If the units digit of b is 9, the units digit of a is 9, but again, the units digits of a and b must be different. It
follows that the units digit of a cannot be the units digit of the square of an integer, so a cannot be the square of an integer.
The answer to the question is definitively no, and Statement (1) is sufficient. Eliminate choices (B), (C), and (E).
Statement (2) precludes a from having 1 as its units digit, but it does nothing else, since the only relevant limitation is that
the two numbers have different units digits. Statement (2) is insufficient.
The correct answer is (A).