# " Help me in solving problem"

• May 31st 2010, 11:18 PM
SEK790
" 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."

(Worried)
• May 31st 2010, 11:32 PM
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Quote:

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."

(Worried)

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

$x\equiv 6\ (\text{mod}\ 1657)$

$x\equiv 5\ (\text{mod}\ 2037)$

1657 and 2035 are pairwise coprime. Solving the CRT gives

$x \equiv 1305722\ (\text{mod}\ 3375309)$

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

$1657\equiv 6\ (\text{mod}\ x)$

$2037\equiv 5\ (\text{mod}\ x)$

$1651\equiv 0\ (\text{mod}\ x)$

$2032\equiv 0\ (\text{mod}\ x)$

where

$1651 = 13\cdot 127$

$2032 = 2^4\cdot 127$