1. Find all solutions, if any, to the system of congruences.
x = 5 (mod 6)
x = 3 (mod 10)
x = 8 (mod 15)
I tried following an example out of my book and got the following, which I'm fairly sure is horribly wrong:
m = 900; M1 = 150; M2 = 90; M3 = 60
150 = a mod 6; a = 12;
90 = b mod 10; b = 20;
60 = c mod 15; c = 30;
z = 5*150*12 + 3*90*20 + 8*60*30
z = 28,800
z = x (mod m)
28,800 = x (mod 900)
so... x = 900. This, doesn't seem right, any help would be appreciated.
2. Use Strong induction to show that every positive integer n can be written as a combination of distinct powers of 2. Ex. .
Here's what I have. I think I have proven it, but I'm not sure if it's by strong induction.
Basis:
Inductive:
Assume:
We must show
if (j+1) is odd. rounded down. y = 0.
/* ie. for ; */
if (j+1) is even. ;
Thus we have shown that the basis and inductive steps are true.