Quote:
If x and y are positive integers, what is the remainder when x is divided by y?
(1) When x is divided by 2y, the remainder is 4
(2) When x + y is divided by y, the remainder is 4
Here's an alternate solution to this question, without using number-plugging:
The question asks us the remainder for x/y
This means, in the expression
x = ky + r . . . (1) (where k is the quotient and r is the remainder)
We have to find the value of r, where 0 =<
r < yAnalyzing St. 1When x is divided by 2y, the remainder is 4That is,
x = (m)(2y) + 4 . . . (2)
Now, upon comparing Equations (1) and (2), can we say r = 4? Let's see:
We know that remainder is always less than the divisor.
Since r is the remainder in Equation 1, we can be sure that
r < ySimilarly, since 4 is the remainder in Equation 2, we can be sure that 4 < 2y. This gives us, y > 2
So, y can be 3, 4, 5, 6, 7 etc.
Now,
if y = 3, then r must be less than 3. So, upon comparing (1) and (2), you'll not say that r = 4. Instead, you'll break down 4 into 3 + 1. So, if y = 3, remainder r = 1
If y = 4, then r must be less than 4. So, upon comparing (1) and (2), you'll not say that r = 4. Instead, you'll write 4 as 4 + 0. So, if y = 4, remainder r = 0
If y > 4, then 4 becomes a valid value for remainder r. So, in this case, upon comparing (1) and (2), we can say that r = 4.
Thus, we see that r can be 0, 1 or 4. Since we have not been able to determine a unique value of r, St. 1 is not sufficient.
Analyzing St. 2When x + y is divided by y, the remainder is 4That is, x + y = qy + 4
Or,
x = (q - 1)y + 4 . . . (3)
Equation (3) conveys that when x is divided by y, the remainder is 4. So, St. 2 is sufficient to find the value of the remainder.
Please note that the way we processed St. 1 was different from the way we processed St. 2, because in St. 1 the divisor was 2y, whereas in St. 2, the divisor was y (the same divisor as in the question)
Hope this alternate solution was useful!
Best Regards
Japinder