Question

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The average of 3 prime numbers lying between 47 to 74 is 191/3, The greatest possible difference between any two out of the 3 prime numbers is​

Answer 1

Please see the attachment

Answer 2

We know that the average of a few integers is the sum of the integers divided by the no. of the integers.

$\displaystyle \text{Average = \frac{\text{Sum of the integers}}{\text{No. of the integers}} }$

Here the average of any three prime numbers between 47 and 74 is 191/3, which is a fraction.

In the average 191/3, the denominator is 3 which is equal to the no. of the prime numbers taken. Thus we can say that the sum of the 3 prime numbers is 191 which is the numerator.

Now, considering the prime numbers lying between 47 and 74,

53, 59, 61, 67, 71, 73

As the three prime numbers are 'between' 47 and 74,  47 won't include among them.

Okay, in how many ways 3 prime numbers from these 6 ones can be taken in such a way that their sum should be 191?

1.  53 + 67 + 71 = 191

2.  59 + 61 + 71 = 191

There are only 2 ways. 73 is not coming in both ways.

Seems that 71, which is the largest prime number among the 5 ones used, is common in both ways. So the greatest possible difference can be found by taking these 5 numbers in ascending order.

Now, taking these 5 prime numbers in ascending order,

53 < 59 < 61 < 67 < 71

From this we get that the least prime number taken is 53. So the greatest possible difference is 71 - 53 = 18.

Hence, 18 is the answer.