Solution: To find the count of integers $ n $ such that $ 1 \leq n \leq 500 $ and $ n \equiv 2 \pmod7 $, we note the sequence starts at 2 and increases by 7 each time: $ 2, 9, 16, \dots $. The general term is $ 7k + 2 $. Solving $ 7k + 2 \leq 500 $ gives $ k \leq rac4987 pprox 71.14 $, so $ k = 0, 1, \dots, 71 $. This yields $ 72 $ integers. oxed72 - Abu Waleed Tea