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

Question Number 124512 by mathocean1 last updated on 03/Dec/20

d e t e r m i n a t e t h e s m a l l e s t i n t e g e r

w h i c h h a s 28 d i v i s o r s .

Commented by mr W last updated on 04/Dec/20

960 ?

Commented by mathocean1 last updated on 04/Dec/20

p l e a s e s i r c a n y o u d e t a i l

Commented by mr W last updated on 04/Dec/20

n = p^{a} q^{b} r^{c} \dots

w i t h p, q, r, \dots = p r i m e = 2, 3, 5, \dots

(a + 1) (b + 1) (c + 1) \dots = 28 = 7 \times 4 = 7 \times 2 \times 2

c a s e 1 : a = 6, b = 3

n_{m i n} = 2^{6} 3^{3} = 1728

c a s e 2 : a = 6, b = 1, c = 1

n_{m i n} = 2^{6} 3^{1} 5^{1} = 960

\Rightarrow n_{m i n} = 960

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