practice problems pending

Q1. Does there exist an integer x satisfying the following conditions?
10x = 1 mod 21
5x = 2 mod 6
4x = 1 mod 7
S1.

No.
Why?
[1] => 
10x = 1 mod 21
=> 10x = 1 mod 3 and 10x = 1 mod 7
=> x = 1 mod 3 and 3x = 1 mod 7 => 36x = 12 mod 7 => x = 5 mod 7
So we have
x = 1 mod 3 and x = 5 mod 7

[2] => 
5x = 2 mod 6
=> 5x = 6k + 2
=> 5x = 2 mod 3 => 2x = 2 mod 3 => 4x = 4 mod 3 => x = 1 mod 3
And
5x = 0 mod 2 => x = 0 mod 2

So we have
x = 1 mod 3
x = 5 mod 7
x = 0 mod 2

[3] =>
4x = 1 mod 7 =>
8x = 2 mod 7 =>
x = 2 mod 7

Which is a contradiction to existing solutions.

Q2. For how many n between 1 and 2520 do we have n^3 = 1 mod 2520?
S2.

2520 = 2^3 * 3^2 * 5 * 7
So
n^3 = 1 mod 8 => n = 1 mod 8
n^3 = 1 mod 9 => n = 1,4,7 mod 9
n^3 = 1 mod 5 => n = 1 mod 5
n^3 = 1 mod 7 => n = 1,2,4 mod 7


For each of those options there is a unique solution (using CRT) between 1 and 2520.
So total 9 valid solutions.

Q3.



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