Thread: " Help me in solving problem"

I did not understand the given problem statement.
Could you please explain it clearly?
And give me step by step answer

The given problem is

" Find the greatest number which on dividing 1657 and 2037 leaves remainders 6 and 5 respectively."

Thanking you , advance.

Please explain in details.

Originally Posted by SEK790

I did not understand the given problem statement.
Could you please explain it clearly?
And give me step by step answer

The given problem is

" Find the greatest number which on dividing 1657 and 2037 leaves remainders 6 and 5 respectively."

Thanking you , advance.

Please explain in details.

Are you sure this problem should be in Pre-Algebra and Algebra? This is a problem for the Chinese Remainder Theorem.





1657 and 2035 are pairwise coprime. Solving the CRT gives



So the answer is 1305722.

EDIT: I misread "greatest number" as "smallest positive integer." There are infinitely many numbers satisfying the given constraints, and there is no greatest number.

EDIT 2: Ah, it seems I made another misinterpretation. I thought you meant "dividing by" instead of "dividing." Okay so we have





This leads to





where





So the answer is 127.

Since you haven't responded, and since the problem is easier than the problem I thought it was at first, I'll write a simpler explanation here.

We are after a number, call it x.

1657 divided by x gives remainder 6. That means that 1657 - 6 = 1651 is a multiple of x.

2037 divided by x gives remainder 5. That means that 2037 - 5 = 2032 is a multiple of x.

We take the greatest common divisor of 1651 and 2032, and to help with that I wrote the prime factorisations in my post above.

If the greatest common divisor had been less than or equal to 6, this problem would have no solution.