Bill owns a large collection of fishing lures consisting of

Analyze the Question:

Bill sold some small, medium, and large lures to his friend. The product of the weights of the lures is 216,000 grams. We know the weights of the individual lures, but we do not know how many of each type he sold. We can factor the product to find the number of 4-gram weights sold. The answer choices are very close together, and these values can help guide the solution process.

Identify the Task:

We will determine the number of 4-gram lures Bill sold to his friend by determining the number of factors of 4 there are in 216,000.

Approach Strategically:

Number properties are the key to solving this problem quickly. Since 3, 4, and 5 do not have any common factors with each other, the number of 4-gram lures Bill sold must be the number of factors of 4 in 216,000. Dividing 216,000 by 4 yields 54,000, and 54,000 is still a multiple of 4 due to its last two digits of 00. Dividing 54,000 by 4 yields 13,500, and 13,500 is still a multiple of 4 due to its last two digits of 00. Dividing 13,500 by 4 yields 3,375, and 3,375 is not a multiple of 4 as its last two digits, 75, are not divisible by 4. That’s three factors of 4 for 216,000, so Bill sold three 4-gram lures to his friend.

The correct answer is Choice (D).

Confirm Your Answer:

We can factor 216,000 completely using the given weights and see that 216,000 = 3^3× 4^3× 5^3. This is the only possible way to get 216,000 from 3, 4, and 5, so we know that three of each of these lures were sold.