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At the end of the T20 cricket match, every player had scored a prime number of runs and the average of the 11 players was also a prime number. No player’s individual runs were the same as anyone else’s or as the average. Nobody had scored more than 45 runs.
Q. What was the minimum number of runs scored by a single player?
- a)5
- b)7
- c)11
- d)13
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
At the end of the T20 cricket match, every player had scored a prime n...
There are 14 prime numbers less than 45 which are as follows:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43
Sum of all these 14 prime numbers = 281
Let’s validate answer options
a) Let the average = 17 ⇒ sum of scores of 11 players = 17 × 11 = 187
187 + the average value (17) = 204
Sum of remaining 2 values to be eliminated = 281 – 204 = 77 (odd number)
= odd value + even value, 2 is the only even prime number value
⇒ odd value must be = 75, not possible
b) Let the average = 19 ⇒ sum of scores of 11 players = 19 × 11 = 209
209 + the average value (19) = 228
Sum of remaining 2 values to be eliminated = 281 – 228 = 53 (odd number)
= odd value + even value, 2 is the only even value among available prime numbers
⇒ odd value must be = 51, not possible
c) Let the average = 23 ⇒ sum of scores of 11 players = 23 × 11 = 253
253 + the average value (23) = 276
Sum of remaining 2 values to be eliminated = 281 – 276 = 5 (odd number)
= odd value + even value = 2 + 3
⇒ Scores of 11 players are 5, 7, 11, 13, 17, 19, 29, 31, 37, 41 and 43 and their average score = 23
d) Let the average = 29 ⇒ sum of scores of 11 players = 29 × 11 = 319 which is more than the sum of all 14 available prime numbers, so not possible
Minimum number of runs scored by single player = 5
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43
Sum of all these 14 prime numbers = 281
Let’s validate answer options
a) Let the average = 17 ⇒ sum of scores of 11 players = 17 × 11 = 187
187 + the average value (17) = 204
Sum of remaining 2 values to be eliminated = 281 – 204 = 77 (odd number)
= odd value + even value, 2 is the only even prime number value
⇒ odd value must be = 75, not possible
b) Let the average = 19 ⇒ sum of scores of 11 players = 19 × 11 = 209
209 + the average value (19) = 228
Sum of remaining 2 values to be eliminated = 281 – 228 = 53 (odd number)
= odd value + even value, 2 is the only even value among available prime numbers
⇒ odd value must be = 51, not possible
c) Let the average = 23 ⇒ sum of scores of 11 players = 23 × 11 = 253
253 + the average value (23) = 276
Sum of remaining 2 values to be eliminated = 281 – 276 = 5 (odd number)
= odd value + even value = 2 + 3
⇒ Scores of 11 players are 5, 7, 11, 13, 17, 19, 29, 31, 37, 41 and 43 and their average score = 23
d) Let the average = 29 ⇒ sum of scores of 11 players = 29 × 11 = 319 which is more than the sum of all 14 available prime numbers, so not possible
Minimum number of runs scored by single player = 5
Most Upvoted Answer
At the end of the T20 cricket match, every player had scored a prime n...
Scored the same number of runs as another player. What is the minimum possible total score of the team?
The smallest prime number greater than 1 is 2, so at least one player must have scored 2 runs. Let's assume that the other 10 players each scored a distinct prime number greater than 2.
The sum of the 10 primes greater than 2 is at least 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 = 158. Adding 2 to this sum gives us a total score of at least 160.
To check if this is a valid scenario, we need to confirm that the average score is also a prime number. The sum of the scores is 160, so the average is 160/11 = 14.54545... which is not a prime number.
Therefore, we need to adjust our assumption and try a different scenario. Let's assume that two players each scored 2 runs, and the other 9 players each scored a distinct prime number greater than 2.
The sum of the 9 primes greater than 2 is at least 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 = 127. Adding 4 to this sum gives us a total score of at least 131.
To check if this is a valid scenario, we need to confirm that the average score is also a prime number. The sum of the scores is 131, so the average is 131/11 = 11.90909... which is not a prime number.
We can continue adjusting our assumption and trying different scenarios, but we will eventually find that there is no solution that satisfies all the conditions. Therefore, there is no minimum possible total score of the team.
The smallest prime number greater than 1 is 2, so at least one player must have scored 2 runs. Let's assume that the other 10 players each scored a distinct prime number greater than 2.
The sum of the 10 primes greater than 2 is at least 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 = 158. Adding 2 to this sum gives us a total score of at least 160.
To check if this is a valid scenario, we need to confirm that the average score is also a prime number. The sum of the scores is 160, so the average is 160/11 = 14.54545... which is not a prime number.
Therefore, we need to adjust our assumption and try a different scenario. Let's assume that two players each scored 2 runs, and the other 9 players each scored a distinct prime number greater than 2.
The sum of the 9 primes greater than 2 is at least 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 = 127. Adding 4 to this sum gives us a total score of at least 131.
To check if this is a valid scenario, we need to confirm that the average score is also a prime number. The sum of the scores is 131, so the average is 131/11 = 11.90909... which is not a prime number.
We can continue adjusting our assumption and trying different scenarios, but we will eventually find that there is no solution that satisfies all the conditions. Therefore, there is no minimum possible total score of the team.
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At the end of the T20 cricket match, every player had scored a prime number of runs and the average of the 11 players was also a prime number. No player’s individual runs were the same as anyone else’s or as the average. Nobody had scored more than 45 runs.Q.What was the minimum number of runs scored by a single player?a)5b)7c)11d)13Correct answer is option 'A'. Can you explain this answer?
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