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Question Number 115011 by joki last updated on 23/Sep/20

given the quadratic function   f(x)=x+2+px+q where p and q are?integer.s  let a,b, and c distinc integers such that  2^(2020)   evenly divides f(a),f(b),and f(c),but  2^(1000)  does not divide b−a and also does not  divide c−a.show that 2^(1021)  just divide b−c?

$${given}\:{the}\:{quadratic}\:{function}\: \\ $$$${f}\left({x}\right)={x}+\mathrm{2}+{px}+{q}\:{where}\:{p}\:{and}\:{q}\:{are}?{integer}.{s} \\ $$$${let}\:{a},{b},\:{and}\:{c}\:{distinc}\:{integers}\:{such}\:{that} \\ $$$$\mathrm{2}^{\mathrm{2020}} \:\:{evenly}\:{divides}\:{f}\left({a}\right),{f}\left({b}\right),{and}\:{f}\left({c}\right),{but} \\ $$$$\mathrm{2}^{\mathrm{1000}} \:{does}\:{not}\:{divide}\:{b}−{a}\:{and}\:{also}\:{does}\:{not} \\ $$$${divide}\:{c}−{a}.{show}\:{that}\:\mathrm{2}^{\mathrm{1021}} \:{just}\:{divide}\:{b}−{c}? \\ $$

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