Try this problem of TIFR GS-2010 using your concepts of number theory and congruence based on natural numbers.
Which of the following statements is false?
NUMBER THEORY
CONGRUENCE
CHINESE REMAINDER THEOREM
Answer:There exists a natural number which when divided by $6$ leaves remainder $2$ and when divided by $9$ leaves remainder $1$
TIFR 2010|PART B |PROBLEM 12
ELEMENTARY NUMBER THEORY DAVID M.BURTON
Let us take the equations $x\equiv1(mod 3)$ and $x\equiv0(mod 4)$
Now we will apply Chinese remainder theorem to get the value of $x$
Since $3$,$4$ are relatively prime,gcd($3$,$4$)$=1$. Let $m=3\times4=12$
Then $M_1=4$,$M_2=3$.
Then gcd($M_1$,$3$)$=1$,gcd($M_2$,$4$)$=1$
Since gcd($4$,$3$)$=1$,therefore the linear congruence equation $4x\equiv1(mod 3)$ has a unique solution and $x\equiv1(mod 3)$ is the solution.
Since gcd($3$,$4$)$=1$,therefore the linear congruence equation $3x\equiv0(mod 4)$ has a solution and $x\equiv4(mod 4)$ is the solution.
Therefore,$x=1\times4\times1 +0\times3\times4=4$ is a solution.
The solution of the given system is $x\equiv4(mod 12)$
So we have used the Chinese Remainder Theorem to check the statements, you may use it to check for other options.
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