A set of 100 open lockers are numbered from 1 - 100. Sarah comes by and closes all the lockers with even numbers. Then Sarah walks past the lockers again and checks the ones numbered with multiples of 3. If the locker is closed, she opens it; if it is open, she closes it. She repeats this with the multiples of 4, 5, 6, … and so on to 100. When she has finished, which lockers will be open? Explain how you know.
all square numbers such as 1,4,9,16,25,36,49,64,81,100,121,....625,676,729,784,841,900,961.
To determine which lockers will be open, we need to understand the pattern of the opening and closing of the lockers by Sarah.
First, Sarah closes all the lockers with even numbers. This means that only the odd-numbered lockers (1, 3, 5, 7, ...) remain open.
Next, Sarah checks the lockers numbered with multiples of 3. For these lockers, Sarah will open the ones that are closed and close the ones that are open. So, at this point, lockers 3, 6, 9, 12, ... will be closed, while lockers 1, 4, 5, 7, 8, 10, 11, 13, ... will be open.
Sarah repeats this process with the multiples of 4, 5, 6, ..., all the way up to 100. Each time she checks the lockers, she will change their status (open to closed or closed to open) based on their previous status.
Let's look at the multiples of each number:
- For locker 4, it will be opened by Sarah on her second pass, but closed again on her fourth pass.
- For locker 5, it will be opened by Sarah on her second pass, but left unchanged on her fifth pass.
- For locker 6, it will be closed by Sarah on her second pass and opened again on her sixth pass.
This pattern continues for each locker. Each locker is affected by Sarah depending on the number of times its multiples appear in the sequence. A locker will be open if it has an odd number of factors after all the passes are completed.
Factors come in pairs, like (1, 4) for locker 4 since 1 and 4 are both factors of 4. So, to have an odd number of factors, a locker must be a perfect square, as squares have an odd number of factors.
Looking at the lockers from 1 to 100, the lockers that will be open are those with perfect square numbers:
1, 4, 9, 16, 25, 36, 49, 64, 81, and 100.
These are the lockers that Sarah will leave open after completing her passes.