Hello everyone, I have a question. During a congruence calculation, I found a very strange pattern.
Let a,t be integers and d the largest factor of t having a power greater than or equal to 2,
If a^2 = 0 (mod t) then a = 0 (mod t/d)
For example :
a^2 = 0 (mod 8) a^2 = 0 (mod 16)
8 = 2^3 Here 2 is the largest factor 16 = 2^4 = 4^2 Here 4 is the largest factor
a = 0 (mod 8/2) = 0 (mod 4) a = 0 (mod 16/4) = 0 (mod 4)
Is that true? And if it is, how can we prove it?
Honestly, I really don't know what to make of that.
Think in term of divisibility!
Let t=k.d² so no square remains in k (as d is the larger factor having a power ≥ 2)
a²≡0 (mod t) means that t ∣ a² => k.d² ∣ a²
As there is no square in k, then k.d ∣ a and, as k.d=t/d, t/d ∣ a
In other words a≡0 (mod t/d) QED
Is all this clear ?