The-sum-of-all-the-divisors-of-24-is-60-Find-the-sum-of-all-the-divisors-of-24-79-

Question Number 202672 by BaliramKumar last updated on 31/Dec/23

$$\mathrm{The}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{all}\:\mathrm{the}\:\mathrm{divisors}\:\mathrm{of}\:\mathrm{24}\:\mathrm{is}\:\mathrm{60}. \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{all}\:\mathrm{the}\:\mathrm{divisors}\:\mathrm{of}\:\mathrm{24}×\mathrm{79}. \\ $$

Answered by a.lgnaoui last updated on 31/Dec/23

$$\mathrm{60} \\ $$

Commented by BaliramKumar last updated on 31/Dec/23

$$\mathrm{sum}\:\mathrm{of}\:\mathrm{divisors}\:\mathrm{of}\:\mathrm{1896}=?\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left[\mathrm{24}×\mathrm{79}\:=\mathrm{1896}\right] \\ $$

Commented by Panav last updated on 31/Dec/23

$$\mathrm{61} \\ $$

Answered by MM42 last updated on 31/Dec/23

d∣24 & Σd=60 d′∣79 & Σd′=80 ⇒Σd′′=4800 ✓ ; d′′∣24×79

$${d}\mid\mathrm{24}\:\:\&\:\:\Sigma{d}=\mathrm{60} \\ $$$${d}'\mid\mathrm{79}\:\:\:\&\:\:\:\Sigma{d}'=\mathrm{80} \\ $$$$\Rightarrow\Sigma{d}''=\mathrm{4800}\:\:\checkmark\:\:\:;\:{d}''\mid\mathrm{24}×\mathrm{79} \\ $$

Commented by BaliramKumar last updated on 31/Dec/23

$$\mathrm{yes} \\ $$

Commented by MM42 last updated on 31/Dec/23

let A={d ;d∣m} & B={d′ ; d′∣n} & C={ d′′ ; d′′∣mn} ;(m,n)=1 ⇒C={1,d_1 ,...,d_m ,d′_1 ,...,d_n ^′ ,d_1 d_1 ^′ ,...,d_m d_n ^′ } ⇒Σd^(′′) =Σd×Σd′

$${let}\:\:{A}=\left\{{d}\:\:\:;{d}\mid{m}\right\}\:\:\&\:{B}=\left\{{d}'\:\:;\:{d}'\mid{n}\right\}\:\:\&\:\:{C}=\left\{\:{d}''\:;\:\:{d}''\mid{mn}\right\}\:;\left({m},{n}\right)=\mathrm{1} \\ $$$$\Rightarrow{C}=\left\{\mathrm{1},{d}_{\mathrm{1}} ,…,{d}_{{m}} ,{d}'_{\mathrm{1}} ,…,{d}_{{n}} ^{'} ,{d}_{\mathrm{1}} {d}_{\mathrm{1}} ^{'} ,…,{d}_{{m}} {d}_{{n}} ^{'} \right\} \\ $$$$\Rightarrow\Sigma{d}^{''} =\Sigma{d}×\Sigma{d}'\:\: \\ $$

Commented by BaliramKumar last updated on 31/Dec/23

$$\mathrm{yes}\:\mathrm{sir} \\ $$

Answered by BaliramKumar last updated on 31/Dec/23

σ_1 (24) = 60 σ_1 (79) = 1+79 = 80 σ_1 (24×79) = 60×80 = 4800 σ_1 (a) = m σ_1 (b) = n σ_1 (ab) = mn Hcf(a, b)=1 & a ≠ b

$$\sigma_{\mathrm{1}} \left(\mathrm{24}\right)\:=\:\mathrm{60} \\ $$$$\sigma_{\mathrm{1}} \left(\mathrm{79}\right)\:=\:\mathrm{1}+\mathrm{79}\:=\:\mathrm{80} \\ $$$$\sigma_{\mathrm{1}} \left(\mathrm{24}×\mathrm{79}\right)\:=\:\mathrm{60}×\mathrm{80}\:=\:\mathrm{4800} \\ $$$$\sigma_{\mathrm{1}} \left(\mathrm{a}\right)\:=\:\mathrm{m}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\sigma_{\mathrm{1}} \left(\mathrm{b}\right)\:=\:\mathrm{n} \\ $$$$\sigma_{\mathrm{1}} \left(\mathrm{ab}\right)\:=\:\mathrm{mn}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Hcf}\left(\mathrm{a},\:\mathrm{b}\right)=\mathrm{1}\:\:\:\&\:\mathrm{a}\:\neq\:\mathrm{b} \\ $$$$ \\ $$

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