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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
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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

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