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natural numbers written a system with base a prime number

Source: IMAC Arhimede 2014 p5

May 6, 2019
number theoryNumerical systemsSystemBinomialProductmoduloprime numbers

Problem Statement

Let pp be a prime number. The natural numbers mm and nn are written in the system with the base pp as n=a0+a1p+...+akpkn = a_0 + a_1p +...+ a_kp^k and m=b0+b1p+..+bkpkm = b_0 + b_1p +..+ b_kp^k. Prove that (nm)≡∏i=0k(aibi)(modp){n \choose m} \equiv \prod_{i=0}^{k}{a_i \choose b_i} (mod p)
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