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

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 .

Facebook Tweet Pin