If a and b are positive integers, what is the units digit of

If a and b are positive integers, what is the units digit of b ?

(1) b is 25% greater than a --> b=1.25a --> b/a = 5/4 --> b is a multiple of 5. So, the units digit of b could be 0 as well as 5. Not sufficient.

(2) If b is decreased by 50%, the result is not an integer --> b/2 is not an integer --> b is not even. Not sufficient.

(1)+(2) Since b is not even, then its units digit cannot be 0, thus from (1) its units digit must be 5. Sufficient.

Answer: C.

Hope it's clear.
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Hi Bunuel,

I think it should be A as the question stem states that a & b are positive integers, ruling out the possibility of b being 0.

Your comments please

Not really. Positive integers can also include numbers such as 10, 20, 35 , 77 or even 1000. Having a ,b POSITIVE INTEGERS only means that the selected integers are >0 NOT that their unit's digit is 0 (this may or may not be true as shown below).

Per statement 1, b = 1.25 a

This statement can not be sufficient as you do not know about a.

Take 2 cases:

a= 4, b = 5, units digit = 5

a =16, b = 20, unit's digit = 0 . Hence you get 2 different answers. Thus this statement is not sufficient.
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Bunuel has mentioned that "the units digit of b could be 0"...he is not saying b is 0..Hope this clears your doubt.

1. b=5a/4
so a must be a multiple of 4, and b a multiple of 5.
a=4, b=5. a=8, b=10, 2 options, not sufficient.

2. b/2 = non-integer. so b is odd.

1+2. b is odd, and b is a multiple of 5.
5, 15, 25, 35, 45, etc. so always b has units digit 5.

Given : positive integers a and b
DS: units digit of b?

Statement 1 : b = 1.25 a = (5/4)a
So a is multiple of 4 and b is a multiple of 5
a= 4, b =5
a=8, b=10
a=12, b=15
a=16, b= 20

So, unit's digit of b is either 0 or 5
NOT SUFFICIENT

Statement 2 : Lets say the resultant value is c. So c = 0.5b and c is not an integer.
So, c must be an odd integer. Unit's digit can be 1,3,5,7,9
NOT SUFFICIENT

Combined : Unit's digit = 5
SUFFICIENT

Do you mean that "b" must be an odd integer? Typo.

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).

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