Deepthi Subbu wrote:If p and n are positive integers and p > n, what is the remainder when p² - n² is divided by 15 ?
(1) The remainder when p + n is divided by 5 is 1.
(2) The remainder when p - n is divided by 3 is 1.
Yes you can select numbers carefully which will save an enormous amount time in this problem. But to do so you must have a good idea about numbers. Choose the numbers such that they satisfy both statement.
Note that for integer values of p and n, the values of (p + n) and (p - n) must be either both even or both odd. If one of them is even and other one is odd, we will get fractional values for p and n. Let's proceed with both odd case.
Statement 1 says, the remainder when p + n is divided by 5 is 1.
As we are considering odd values of (p + n), (p + n) can be 11, 21, 31 etc
Statement 2 says, the remainder when (p - n) is divided by 3 is 1.
As we are considering odd values of (p - n), (p - n) can be 1, 7, 13, 17 etc
Now, note that for each possible value of (p + n), we can always choose (p - n) to be equal to 1. For example, (p + n) = (6 + 5) = 11 => (p - n) = (6 - 5) = 1. Same goes for other values.
This helps us to conclude that for each possible values of (p + n), the value of (p² - n²) = (p - n)(p + n) can be equal to the value of (p + n) itself. As (p - n) always can be equal to 1.
Now some of the possible values of (p² - n²) are 1, 21, 31 etc.
Each of them gives different remainder when divided by 15.
Hence we can conclude that both the statement together is also not sufficient to answer the question as our numbers satisfies all the conditions provided.
The correct answer is .
