The average of three consecutive prime numbers is 223/3. What is the difference between the greatest and the smallest number that can be a part of such a set?

Question

Answer

8

Answer

Solution:

Step 1: Identify the Consecutive Primes

Let's assume that the three consecutive prime numbers are x, x+2, and x+4.

Step 2: Find the Average

The average of these three numbers is given as 223/3.

Therefore, we can write:

(x + x + 2 + x + 4)/3 = 223/3

Simplifying this equation, we get:

3x + 6 = 223

3x = 217

x = 73

Therefore, the three consecutive prime numbers are 73, 75, and 77.

Step 3: Find the Difference

The greatest number in this set is 77 and the smallest number is 73.

Therefore, the difference between the greatest and smallest number is:

77 - 73 = 4

Step 4: Verify that the Numbers are Prime

To verify that these numbers are indeed prime, we can check their divisibility by all numbers from 2 to the square root of the number.

In this case, we can check that:

- 73 is prime because it is not divisible by 2, 3, 5, 7, 11, or 13.
- 75 is not prime because it is divisible by 3 and 5.
- 77 is not prime because it is divisible by 7.

Therefore, the only prime number in this set is 73.

Similar GATE Doubts

sum of eight consecutive odd numbers is 656. The average of four consecutive evennumbers is 87. What is the sum of the smallest odd number and second largest even number? Correct answer is '163'. Can you explain this answer?

The sum of eight consecutive odd numbers is 656. The average of four consecutive even numbers is 87. What is the sum of the smallest odd number and second largest even number?

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