Question Number 124107 by mathocean1 last updated on 30/Nov/20
$${n}\:\in\:\mathbb{N}\:{and}\:{p}\:{is}\:{a}\:{prime}\:{number}\:\left({p}\geqslant\mathrm{3}\right). \\ $$$${a}\:{and}\:{b}\:{are}\:{defined}\:{by}:\:{a}=\mathrm{2}^{{n}} \:{and} \\ $$$${b}={a}×{b}.\:\:{S}\left({a}\right)\:{is}\:{the}\:{sum}\:{of}\:{divisors}\: \\ $$$${of}\:{a}\:{and}\:{S}\left({b}\right)\:\:{is}\:{the}\:{sum}\:{of}\:{divisors} \\ $$$${of}\:{b}. \\ $$$$\mathrm{1}.{Determinate}\:{the}\:{set}\:{of}\:{divisors}\:{of} \\ $$$${a}\:{and}\:{the}\:{set}\:{of}\:{divisors}\:{of}\:{b}. \\ $$$$\mathrm{2}.{show}\:{that}\:{S}\left({a}\right)+\mathrm{2}^{{n}+\mathrm{1}} =\mathrm{1}+\mathrm{2}{S}\left({a}\right) \\ $$$${then}\:{calculate}\:{S}\left({a}\right). \\ $$$$\mathrm{3}.\:{write}\:{S}\left({b}\right)\:{in}\:{function}\:{of}\:{S}\left({a}\right)\:{then} \\ $$$${calculate}\:{S}\left({b}\right). \\ $$