Decimal digits $a,b,c$ satisfy \[ 37\mid (a0a0\ldots a0b0c0c\ldots 0c)_{10} \] where there are $1001$ a's and $1001$ c's. Prove that $b=a+c$.
Source: Lithuania NMO 2010
Tags: modular arithmetic, number theory proposed, number theory
Decimal digits $a,b,c$ satisfy \[ 37\mid (a0a0\ldots a0b0c0c\ldots 0c)_{10} \] where there are $1001$ a's and $1001$ c's. Prove that $b=a+c$.