Yes Sir @TerryTao. Thanks a lot. Now I know the answer to my question.

Thanks Sir @TerryTao, I should have make the conditions for p and q more tight. That is I should have included that p and q must not be divisible by 10. Is it okay Sir to ask if the statement: There are infinitely many integers p and q and t satisfying the system for all j and k, being TRUE. Thanks again Sir.

Can I also post this in mathematics stack exchange? Thanks a lot.

That is true @JohnBinder. Those cases are exceptions.

Thanks @GerhardPaseman will try to do that. Thanks again. But if ever you have the result, kindly inform me Sir.

Thanks @GerhardPaseman, sorry if I am having a hard time on understanding your comment, In particular what does $m$ represent and what is the meaning of combining the two system. As well as pt+j...qt+k...qt+k...pt+j. I am hopeful that with your help I can understand the answer to my query. Thanks again.

Yes Sir @GerryMyerson, it must be that p and q are relatively prime.