[Refreshing an old question] Find the smallest number which is a multiple of 75 and has 75 divisors.


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Question Number 38933 by MrW3 last updated on 01/Jul/18

$[R e f r e s h i n g a n o l d q u e s t i o n]$ $F i n d t h e s m a l l e s t n u m b e r w h i c h i s$ $a m u l t i p l e o f 75 a n d h a s 75 d i v i s o r s .$
Commented by math1967 last updated on 01/Jul/18

$75 = 3 \times 5^{2} ∴ f a c t o r o f 75$ $(1 + 1) (2 + 1) - 1 = 5$ $l e t 3^{x} \times 5^{y} h a s 75 f a c t o r$ $(x + 1) (y + 1) = 76 = 19 \times 4$ $f o r s m a l l e s t m u l t i p l e o f 75$ $x = 18 y = 3 ∴ r e q u i r e d n u m b e r 3^{18} \times 5^{3}$ $I s i t c o r r e c t ?$
Commented by MrW3 last updated on 01/Jul/18

$N o t y e t, b u t y o u a r e a p p r o a c h i n g t h e$ $g o a l . W h y 76 ? T h e r e q u e s t e d n u m b e r$ $s h o u l d h a v e e x a c t l y 75 f a c t o r s . B e s i d e s$ $a n o t h e r h i n t : 2 \times 75 i s s m a l l e r t h a n$ $3 \times 75, a n d b o t h a r e a m u l t i p l e o f 75!$
Commented by MrW3 last updated on 01/Jul/18

$H i n t : 75 h a s 6 f a c t o r s : 1, 3, 5, 15, 25, 75 .$
Commented by math1967 last updated on 01/Jul/18

$(x + 1) (y + 1) - 1 = 75$ $∴ (x + 1) (y + 1) = 76$
Commented by math1967 last updated on 01/Jul/18

$(x + 1) (y + 1) = 75$ $x = 24 y = 2$ $3^{24} \times 5^{2} p l s c h e c k$
Commented by MrW3 last updated on 01/Jul/18

$n o t c o r r e c t s i r .$
Commented by math1967 last updated on 01/Jul/18

$2^{4} \times 3^{3} \times 75 = 32400 ?$
Commented by MrW3 last updated on 01/Jul/18

${t h a t}^{'} s c o r r e c t s i r .$
Commented by MrW3 last updated on 01/Jul/18

$F o r m o r e s e e o r i g i n a l q u e s t i o n i n$ $Q 11075 .$
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