Sun Tzi's Modular Problem

[Guest]

x=1 mod (3)
x=3 mod (5)
x=5 mod (7)

Find the smallest positive solution and find the family of solutions

I have M=105, M1=35, M2=21, M3=15

But i always come out with 173 as the answer, and the smallest solution being 68, which is wrong...

I am not really sure what family of solutions means as well...

 

[pka]

Is your post written correctly?
If so, the smallest solution is 103.
Note that mod(68,3)=2 not 1.

 

[galactus]

My modular arithmetic is a little rusty, so bear with me and check my answer.

\(\displaystyle P_{1}: x=1mod3\)

\(\displaystyle P_{2}: x=3mod5\)

\(\displaystyle P_{3}: x=5 mod7\)


In \(\displaystyle P_{1}\), it is the same as \(\displaystyle x=3t+1\)

Sub into \(\displaystyle P_{2}\)

\(\displaystyle 3t+1=3mod5\)

\(\displaystyle 3t=2mod5\)

reduces to:

\(\displaystyle P_{4}: t=4mod5\)

\(\displaystyle t=5p+4\)

Sub into x:

\(\displaystyle x=3(5p+4)+1=15p+13\)

Sub into \(\displaystyle P_{3}\)

\(\displaystyle 15p+13=5mod7\)

(Here's a cool trick that can be handy)Cast out 7's:

\(\displaystyle s=-1mod7\)

\(\displaystyle s=7z-1\)

Sub into \(\displaystyle 15(7z-1)+13=5mod7\)

\(\displaystyle 105z-2\)

This is the family of solutions.

\(\displaystyle {-2, 103, 208, 313, 418,........................}\)