honchos wrote:
If a and b are positive integers, what is the units digit of b ?
(1) b is 25% greater than a.
(2) If b is decreased by 50%, the result is not an integer.
OE
You can use smart numbers or algebraic theory to solve this problem; both approaches are shown below. (You can also mix and match for the different statements.)
Smart Numbers
(1) NOT SUFFICIENT: 25% of a must be an integer, so choose multiples of 4 for a. If a = 4, then b = 5, with units digit 5. If a = 8, then b = 10, with units digit 0. There are at least two possible values for the units digit of b.
(2) NOT SUFFICIENT: If b = 3, then 50% of b is indeed not an integer (1.5). If, on the other hand, b = 4, then 50% of b is still an integer (2). Any even value for b will still result in an integer, so b can’t be even; b must be odd. The units digit of b could be any odd digit (1, 3, 5, 7, or 9).
(1) AND (2) SUFFICIENT: Statement 2 limits the units digit of b to 1, 3, 5, 7, or 9. Statement 1 requires a to be a multiple of 4. If a = 4, then b = 5, with units digit 5. This time, a cannot equal 8, because then b will be 10, but a units digit of 0 isn’t allowed. The next possible value is a =12, in which case b = 15, with units digit 5. Is this a pattern?
Yes, if you try the next couple of values, the only acceptable values will result in b having a units digit of 5. The two statements together are sufficient.
Algebraic Theory
(1) NOT SUFFICIENT: According to this statement, b = a + 0.25a = 1.25a, or b = (5/4)a. In other words, if a is divided by 4 and multiplied by 5, the result is b. Since both a and b are integers, a must be a multiple of 4, and b must be a multiple of 5. The units digit of b must be either 0 or 5; there are still two possibilities, so the statement is insufficient.
(2) NOT SUFFICIENT:
According to this statement, b / 2 is not an integer. This is true only when integer b is odd, which narrows the possibilities, but there are still 5 possibilities (1, 3, 5, 7, or 9).
(1) AND (2) SUFFICIENT:
Statement (1) can only be satisfied if the units digit of b is 0 or 5, and statement (2) only if the units digit is 1, 3, 5, 7, or 9. The only common digit is 5, so, if both statements are true, the units digit of b must be 5.
The correct answer is (C).