### Instruction: The sequence 1,000,000; 500,000; 250,000 and so on, is made by repeatedly dividing by 2. What is the last integer in this sequence?### Response: I notice that the sequence is a geometric sequence, where each term is half of the previous term.
This means that the ratio of any two consecutive terms is 1/2.
I wonder how many times I need to divide by 2 to reach the last integer in the sequence.
If I divide by 2 n times, then the n-th term of the sequence is 1,000,000 / 2^n.
I want to find the smallest value of n such that 1,000,000 / 2^n is an integer.
Equivalently, I want to find the largest power of 2 that divides 1,000,000 exactly.
I recall that the prime factorization of 1,000,000 is 2^6 * 5^6.
This means that the largest power of 2 that divides 1,000,000 is 2^6, or 64.
This means that I need to divide by 2 six times to reach the last integer in the sequence.