John counts up from 1 to 13, and then immediately counts down again to 1, and then back up to 13, and so on, alternately counting up and down: What is the \(1234th\) integer in his list?
+23245 As you count from 1 to 13 and then back down again, you get this set of numbers:
1-2-3-4-5-6-7-8-9-10-11-12-13-12-11-10-9-8-7-6-5-4-3-2
Then your next set of numbers repeat the above list
1-2-3-4- etc.
There are 24 numbers in each list.
So, how many sets of 24 are in 1234?
Dividing 1234 by 24, we get a quotient of 51 and a remainder of 10.
The quotient of 51 tells me that there are 51 groups of 24 numbers (which I really don't care about)
and the remainder of 10 tells me how far to keep counting (I care about this!)
So what number is the tenth number in the list
1-2-3-4-5-6-7-8-9-10-11-12-13-12-11-10-9-8-7-6-5-4-3-2 ?
The tenth number in the list is my answer ...
We can note that each repetition of the sequence [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2] has length 24. We do not include the extra 1 at the end to prevent double counting. If we divide 1234 by 24, we get 51 with some remainder. Multiplying 24 by 51, we get 1224. Subtracting this from 1234, we get a result of 10. Thus, our answer is the 10th number in the list: 10.
