Suelyn counts up from 1 to 9, and then immediately counts down again to 1, and then back up to 9, and so on, alternately counting up and down $$(1, 2, 3,4,5,6,7,8,9,8,7,6,5,4,3,2,1,2,3,4,\ldots ).$$ What is the $1000^{\text{th}}$ integer in her list?
I notice that Suelyn's list has a repeating pattern of 16 integers: $$(1, 2, 3,4,5,6,7,8,9,8,7,6,5,4,3,2).$$ So, to find the $1000^{\text{th}}$ integer, I can divide 1000 by 16 and look at the remainder. 1000 divided by 16 is 62 with a remainder of 8. That means that the $1000^{\text{th}}$ integer is the same as the $8^{\text{th}}$ integer in the pattern. The $8^{\text{th}}$ integer in the pattern is 8. Therefore, the $1000^{\text{th}}$ integer in Suelyn's list is 8.

