Final answer:
To find the number of integers in Cameron's list, identify the smallest positive multiple of 20 that is a perfect square and the smallest positive multiple of 20 that is a perfect cube. They are 100 and 8000, respectively. Counting all multiples of 20 between them, including 100 and 8000, gives us 396 integers.
Step-by-step explanation:
To solve the problem, we first need to find the smallest positive multiple of 20 that is a perfect square. A perfect square is a number that can be expressed as the square of an integer. The prime factorization of 20 is 22 × 5. Therefore, to make it a perfect square, we need at least two 5s, giving us 52. Multiplying this with the 22 we already have in 20, we get 22 × 52, which is 100. This is the smallest multiple of 20 that is also a perfect square.
Next, we need to find the smallest multiple of 20 that is a perfect cube. A perfect cube is a number that can be expressed as the cube of an integer. Again looking at the prime factors of 20, we need three 2s and three 5s to make the smallest cube. So we need 23 × 53, which is 8000. This is the smallest multiple of 20 that is also a perfect cube.
Now, Cameron's list will include all multiples of 20 between 100 and 8000. To find out how many multiples of 20 there are between these two numbers, we divide the larger number by 20 and then subtract the result of dividing the smaller number by 20: (8000 / 20) - (100 / 20) = 400 - 5 = 395. However, since we need to include both the first and the last term in the list, we add one back giving us a total of 396.
Thus, there are 396 integers in Cameron's list.