If the positive integer n is added to each of the integers 6

Hi AbdurRakib,

This question is built around some Number Property logic and requires a bit of 'brute force' to get to the correct answer quickly.

We're told that N is a positive integer. We're asked for the value of N.

(1) 69 + n and 94 + n are the squares of two consecutive integers

This Fact tells us that (69+N) and (94+N) are two CONSECUTIVE PERFECT SQUARES. I'm going to list out some perfect squares so that we can look for a pattern:

1
4
9
16
25
36

Notice as you go down the list that the difference between any two consecutive squares INCREASES (that's the Number Property pattern I referenced earlier). The two numbers that we're told to consider differ by 25 (since 94-69 = 25), so we're looking for two consecutive squares that differ by 25. From the above pattern, we know that there will be JUST ONE pair that differ by 25 (the pairs that come before it will differ by less than 25 and the pairs that come after it will differ by more than 25) - whether we define it or not.

For the record, the numbers are 144 and 169, which makes X=75.
Fact 1 is SUFFICIENT

2) 94 + n and 121 + n are the squares of two consecutive integers.

The same logic that applies to Fact 1 applies to Fact 2, so we don't have to do any more work here.
Fact 2 is SUFFICIENT.

Final Answer: D

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