If x is 1 mod 3 0 mod 5 and 0 mod 11 what is it in mod 165 b

If x is 1 mod 3, 0 mod 5, and 0 mod 11, what is it in mod 165? (b) If y is 0 mod 3, 1 mod 5, and 0 mod 11, what is it in mod 165? (c) If z is 0 mod 3, 0 mod 5, and 1 mod 11, what is it in mod 165? (d) Compute 2x + 4y + 8z, answering in mod 165.

Solution

x=1 mod 3

x=0 mod 5

x=0 mod 11

Hence, x=0 mod 55

x=55m

x=1 mod 3

55m=1 mod 3

m=1 mod 3

m=3n+1

x=55m=55(3n+1)=165n+55

HEnce, x=55 mod 165

b)

x=0 mod 3, x=0 mod 11

So, x=0 mod 33

x=33 m

x=1 mod 5

33m=1 mod 5

3m=1 mod 5

6m=2 mod 5

m=2 mod 5

m=5n+2

x=33m=33(5n+2)=165n+66

x=66 mod 165

c)

x=0 mod 3, x=0 mod 5

x=0 mod 15

x=15m

x=1 mod 11

15m=1 mod 11

4m=1 mod 11

3*4m=3 mod 11

12m=m=3 mod 11

m=11n+3

x=15m=15(11n+3)=165n+45

x=45 mod 165

d)

2x+4y+8z=2*55+4*66+8*45=734=74 mod 165