Problem Solving

yumi2012 wrote:If s and t are positive integers such that s/t = 64.12, which of the following could be the remainder when s is divided by t ?

(A) 2
(B) 4
(C) 8
(D) 20
(E) 45

This problem will be best solved using the remainder formula. Let's first state the remainder formula:

When positive integer x is divided by positive integer y, if integer Q is the quotient and r is the remainder, then x/y = Q + r/y.

In this problem we are given the following:

s/t = 64.12

We can simplify this to read as the remainder formula:

s/t = 64 + 0.12

s/t = 64 + 0.12

s/t = 64 + 12/100

s/t = 64 + 3/25

Because Q is always an integer, we see that Q must be 64, and thus the remainder r/y must be 3/25. We can now equate r/y to 3/25 and determine a possible value for r.

r/y = 3/25

Note that some equivalent values for r/y could be 6/50 or 9/75 or 12/100, and so forth. Note that in all cases, the value of r is a multiple of 3.

Of the answer choices, the only multiple of 3 is 45, so that is a possible value of r.

Answer: E

Hi All,

We’re told that S and T are both POSITIVE INTEGERS and that S/T = 64.12; we’re asked which of the 5 answers COULD be the remainder when S is divided by T. The wording of the question implies that there is more than one possible answer, so the first example that we come up with might not be listed among the 5 choices – and we’ll likely need to do a little extra work to get to the final answer.

This question can be solved with some standard Arithmetic rules and TESTing VALUES (to define the pattern behind this question).

Since S and T are POSITIVE INTEGERS, there is clearly a relationship between those two values: S is equal to 64.12 times T. The ‘quirky’ part of that equation is the .12, so we can start with the simplest example that we know would give us those two decimal points:

S = 6412 and T = 100…. S/T = 6412/100 = 64.12

In this example, the remainder would be 12. This is clearly not among the choices (but that’s not a surprise; remember the question asked what COULD be the remainder, NOT “what IS the remainder”…).

We can ‘reduce’ this fraction and still keep the 64.12 ‘relationship’; since both the numerator and denominator are EVEN numbers, we can divide both by 2…

S = 3206 and T = 50

Here, the remainder would be 6. This too is not among the answer choices. Both S and T are still even though, so we can divide both by 2 again…

S = 1603 and T = 25

This would give us a remainder of 3 – and while we still don’t have a result that’s among the 5 choices, we do have enough information to define the pattern involved in all three answers: they are ALL multiples of 3…. And there’s only one answer among the 5 that fits THAT pattern.

Final Answer: E

While it’s not necessary to do the extra math… if you wanted to, then you could prove that that Answer is correct by using the ‘reducing’ rule in reverse: you can multiply both the numerator and the denominator by the same number and the 64.12 relationship would stay the same.

If you multiply both 1603 and 25 by 15, you will end up with some larger numbers (S = 24045 and T = 375) and the remainder would match the correct answer.

GMAT assassins aren't born, they're made,
Rich