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The Chinese Remainder Theorem

The Chinese Remainder TheoremConsiderOystein Ore mentions a puzzle with a dramatic element from Brahma-Sphuta-Siddhanta (Brahma's Correct System) by Brahmagupta (born 598 AD):An old woman goes to market and a horse steps on her basket and crushes the eggs. The rider offers to pay for the damages and asks her how many eggs she had brought. She does not remember the exact number, but when she had taken them out two at a time, there was one egg left. The same happened when she picked them out three, four, five, and six at a time, but when she took them seven at a time they came out even. What is the smallest number of eggs she could have had? The Chinese Remainder TheoremTheorem: Suppose that m1, m2, .., mr are pairwise relatively prime positive integers, and let a1, a2, .., ar be integers. Then the system of congruences, x ai (mod mi) for 1 i r, has a unique solution modulo M = m1 m2.

Find the smallest multiple of 10 which has remainder 2 when divided by 3, and remainder 3 when divided by 7. We are looking for a number which satisfies the congruences, x ≡ 2 mod 3, x ≡ 3 mod 7, x ≡ 0 mod 2 and x ≡ 0 mod 5. Since, 2, 3, 5 and 7 are all relatively prime in pairs, the Chinese Remainder Theorem tells us that

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Linear Codes

www-math.ucdenver.edu

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