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determinate-the-smallest-integer-which-has-28-divisors-




Question Number 124512 by mathocean1 last updated on 03/Dec/20
determinate the smallest integer  which has 28 divisors.
$${determinate}\:{the}\:{smallest}\:{integer} \\ $$$${which}\:{has}\:\mathrm{28}\:{divisors}. \\ $$
Commented by mr W last updated on 04/Dec/20
960?
$$\mathrm{960}? \\ $$
Commented by mathocean1 last updated on 04/Dec/20
please sir can you detail
$${please}\:{sir}\:{can}\:{you}\:{detail} \\ $$
Commented by mr W last updated on 04/Dec/20
n=p^a q^b r^c ...  with p,q,r,...=prime=2,3,5,...  (a+1)(b+1)(c+1)...=28=7×4=7×2×2  case 1: a=6, b=3  n_(min) =2^6 3^3 =1728  case 2: a=6, b=1, c=1  n_(min) =2^6 3^1 5^1 =960    ⇒n_(min) =960
$${n}={p}^{{a}} {q}^{{b}} {r}^{{c}} … \\ $$$${with}\:{p},{q},{r},…={prime}=\mathrm{2},\mathrm{3},\mathrm{5},… \\ $$$$\left({a}+\mathrm{1}\right)\left({b}+\mathrm{1}\right)\left({c}+\mathrm{1}\right)…=\mathrm{28}=\mathrm{7}×\mathrm{4}=\mathrm{7}×\mathrm{2}×\mathrm{2} \\ $$$${case}\:\mathrm{1}:\:{a}=\mathrm{6},\:{b}=\mathrm{3} \\ $$$${n}_{{min}} =\mathrm{2}^{\mathrm{6}} \mathrm{3}^{\mathrm{3}} =\mathrm{1728} \\ $$$${case}\:\mathrm{2}:\:{a}=\mathrm{6},\:{b}=\mathrm{1},\:{c}=\mathrm{1} \\ $$$${n}_{{min}} =\mathrm{2}^{\mathrm{6}} \mathrm{3}^{\mathrm{1}} \mathrm{5}^{\mathrm{1}} =\mathrm{960} \\ $$$$ \\ $$$$\Rightarrow{n}_{{min}} =\mathrm{960} \\ $$

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