Question

A skeet shooting competition awards prizes for each round as follows:
the first-place winner receives 11 points, the second-place winner receives 7 points, the third place finisher receives 5 points, and the fourth-place finisher receives 2 points. No other prizes are awarded. John competes in several rounds of the skeet shooting competition and receives points every time he competes. If the product of all of the points he receives equals 84,700, in how many rounds does he participate ?

• A. $5$
• B. $7$
• C. $6$
• D. $8$

Answer 1

The correct option is B $7$

7 rounds

Notice that the values for scoring first, second, third, and fourth place in the competition are all prime numbers. Notice also that the product of all of the scores John received is known. Therefore, if you simply take the prime factorization of the product of his scores, you can determine what scores he received (and how many scores he received):

$84,700=847\times 100=7\times 121\times 2\times 2\times 5\times 5=7\times 11\times 11\times 2\times 2\times 5\times 5$

Thus, John received first place twice (11 points each), second place once (7 points each), third place twice (5 points each), and fourth place twice (2 points each.) He received a prize 7 times, so he competed in 7 rounds.