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55th in the list (Posted on 2010-10-28) Difficulty: 2 of 5
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?

No Solution Yet Submitted by Ady TZIDON    
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Solution solution | Comment 2 of 6 |

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
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