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.