Cameron writes down the smallest positive multiple of 20 that is a perfect square, the smallest positive multiple of 20 that is a perfect cube, and all the multiples of 20 between them.  How many integers are in Cameron's list?
A perfect square that is a multiple of $20 = 2^2 \cdot 5^1$ must be a multiple of $2^2 \cdot 5^2 = 100$.  A perfect cube that is a multiple of 20 must be a multiple of $2^3 \cdot 5^3 = 1000$.  Our goal is thus to count the multiples of 20 from 100 to 1000 inclusive: $$ 100 \le 20n \le 1000. $$Dividing this entire inequality by 20 we get $5 \le n \le 50$, so there are $50 - 5 + 1 = \boxed{46}$ integers in Cameron's list.
46
Level 5
Number Theory
Problem: Cameron writes down the smallest positive multiple of 20 that is a perfect square, the smallest positive multiple of 20 that is a perfect cube, and all the multiples of 20 between them.  How many integers are in Cameron's list?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the smallest positive multiple of 20 that is a perfect square, we start by determining the prime factorization of 20, which is 2^2 * 5^1. For a number to be a perfect square, all prime factors must have even exponents. The smallest perfect square multiple of 20 must therefore have the form 2^2 * 5^2, resulting in 100, since we need to increase the exponent of 5 to the next even number.

Next, we find the smallest positive multiple of 20 that is a perfect cube. For a number to be a perfect cube, all prime factors must have exponents that are multiples of 3. The prime factorization of 20 is again 2^2 * 5^1. To make the exponents multiples of 3, we need to increase the exponent of 2 to 3 and the exponent of 5 to 3. Thus, the smallest perfect cube multiple of 20 is 2^3 * 5^3, which equals 1000.

Now, we identify all multiples of 20 between 100 and 1000. The multiples of 20 can be expressed as 20n, where n is a positive integer. The smallest integer n that gives a multiple of 20 equal to or greater than 100 is n = 5 (since 20 * 5 = 100). The largest integer n that gives a multiple of 20 less than or equal to 1000 is n = 50 (since 20 * 50 = 1000).

To find the total count of integers in Cameron's list, we calculate the number of integers n from 5 to 50, inclusive. This is done by subtracting the smallest n from the largest n and adding one (to include both endpoints): 

50 - 5 + 1 = 46.

Thus, the final answer is 46.<|END_OF_LATENT|> The final answer is $46$.