1. The sequence 2, 3, 5, 6, 7, 10, 11, $\ldots$ contains all the positive integers from least to greatest that are neither squares nor cubes. What is the $400^{\mathrm{th}}$ term of the sequence?

2. The first digit of a string of 2002 digits is a 1. Any two-digit number formed by consecutive digits within this string is divisible by 19 or 31. What is the largest possible last digit in this string?

Guest Jun 3, 2018

#1**0 **

1) First list ALL consecutive numbers from 2 to the 400th term, which is **401. **Then subtract all the squares of all numbers 2 to 20, which come to **19** numbers that are squares. Then subtract all the cubes of all numbers from 2 to 7, which come** 6 **numbers that are cubes.

So, you have: 401 - 19 - 6 =376 terms ending in 401. Then replace the 25 square and cubes with 25 numbers added to 401. Or: 401 + 25 =**426, which should be the 400th term of your sequence.**

**Note: All the numbers from 401 to 426 contain no squares or cubes.**

Guest Jun 3, 2018