Bo'linuvchanlik. Sonning natural bo'luvchilar soni va yig'indisi bo‘yicha savol

$240$ sonini tub ko‘paytuvchilarga ajratamiz: $240 = 2^{4}\cdot 3^{1}\cdot 5^{1}$. Agar $n=p_{1}^{a_{1}}p_{2}^{a_{2}}\dots p_{k}^{a_{k}}$ bo‘lsa, $n$ ning barcha natural bo‘luvchilarining yig‘indisi quyidagi formulaga teng: \[ \sigma (n)=\prod_{i=1}^{k}\frac{p_{i}^{\,a_{i}+1}-1}{p_{i}-1}. \] Shu formulani $240$ ga qo‘llaymiz: \[ \sigma (240)=\frac{2^{4+1}-1}{2-1}\cdot\frac{3^{1+1}-1}{3-1}\cdot\frac{5^{1+1}-1}{5-1} =\frac{32-1}{1}\cdot\frac{9-1}{2}\cdot\frac{25-1}{4} =31\cdot4\cdot6=744. \] Demak, $240$ sonining barcha natural bo‘luvchilarining yig‘indisi $\boxed{744}$, ya’ni variant **A**.