List all the numbers below 2010 having all the following features:
1. n-1 is a multiple of 3.
2. n^2-1 is a multiple of 8.
3. n^3-1 is a multiple of 10.
If you arrange this list in an ascending order, what number will be in the 55th place?
The first number on the list would be 1.
Then, n of course has a cycle of 3 when considered mod 3.
n^2 has a cycle of 4 mod 8: 1,4,1,0,..., but for the current consideration, the fact that every other one is the digit 1 is all that matters, so we can consider the cycle being 2 for the remainder 1.
n^3 mod 10 has a cycle length of 10.
The LCM of 3, 2 and 10 is 30, so starting with 1, every 30th number is on the list as being congruent to 1 in all the pertinent moduli.
If the first is 1, then we need to go 54 more to get to the 55th:
1 + 54*30 = 1621, which is the answer.
Posted by Charlie
on 2010-10-29 01:25:53