harsh.champ wrote:Three prime numbers are chosen by Sanjeev such that each chosen prime number is greater than 115 but less than 155. The sum of the three chosen numbers is 407. Find the difference between the smallest and the largest of the chosen numbers.
1. The difference between the smallest and the largest of the chosen numbers is less than 5.
2. The difference between the smallest and the largest of the chosen numbers is greater than 8
This is not a DS question; for one thing the Statements are obviously contradictory, and for another, Condition 1 is mathematically impossible. It's really two different Problem Solving questions. It's also not a very GMAT-like question, since you need to work with lists of three-digit prime numbers, something not convenient to do quickly. There are better questions to use for practice. In any event, if anyone's interested, it can be solved:
First, if three numbers add to 407, at least one must be below or equal to the average 407/3 = 135 2/3, and at least one must be above or equal to the average 135 2/3. Since our numbers are prime integers, at least one must be less than 135 and one must be greater than 135.
Condition 1 could only be true if two of the primes are equal; if we have three large primes, they must be odd, and they cannot be p, p+2, p+4, since among any three consecutive odd numbers, one must be a multiple of 3 (and therefore not prime if the numbers are larger than 3). Since the primes would need to be very close together if the largest and smallest are less than five apart, they would all need to be very close to the average of 135 2/3, with at least one prime below this average and one prime above this average. Now, the nearest prime to 135 which is less than 135 is 131, and there is no prime larger than 131 but less than 136, so it is impossible to find three primes that satisfy this condition. I can only guess that the question designer intended the solution to be 133, 137, 137, but 133 is not prime (it's divisible by 7 and 19).
For Condition 2, the difference between the smallest and largest primes must be at least 10 (since primes this size are all odd). There aren't many primes to choose from in the given range:
127, 131, 137, 139, 149, 151
The smallest prime must be below average; it must be either 127 or 131. If the smallest is 127, we need two primes which add to 280, or average to 140; only 131 and 149 work. If the smallest is 131, then we need two primes which add to 276, or average to 138; while 137 and 139 work, then our list becomes 131, 137, 139, and our primes are too close together. So only the list 127, 131, 149 works, and the difference between the largest and smallest is 22.
