How many ways can you express 30,030 as the product of 4 positive numbers? (excluding 1)


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Question Number 5796 by FilupSmith last updated on 28/May/16

$How many ways can you express$ $30, 030 as the product of 4 positive numbers ?$ $(excluding 1)$
Commented by Rasheed Soomro last updated on 28/May/16

$I {d i d n}^{'} t t h i n k k, l, m, n t o b e n e c e s s a r i l y p r i m e!$
Commented by FilupSmith last updated on 28/May/16

$o k s o$ $x as an integer can be split into a maximum$ $amount of products as its prime factors .$ $eg . 100 = 2^{2} 5^{2} = 2 \times 2 \times 5 \times 5$ $so 100 can be split to a \max of 4 numbers$ $the number of ways ylu can express 100$ $as 4 numbers is 4! = 24 ways$ $100 = 2 \times 2 \times 5 \times 5 = 5 \times 5 \times 2 \times 2 = . . . etc$ $to look at combinatioms of 3 it is^{4} C_{3}$ $P L U S the number of ways you can turn$ $2 \times 2 \times 5 \times 5 into 3 numbers .$ $i . e . 2 \times 2 \times 5 \times 5 :$ $a_{1} = 4 \times 5 \times 5 a_{2} = 5 \times 4 \times 5 a_{3} = . . .$ $a_{4} = 2 \times 10 \times 2 . . . e t c a_{n} = . . .$ $the number of ways you can do that = n$ $∴ combinations of 3 =^{4} C_{3} + n$ $t h o u g h t s o r s i m p l i f i c a t i o n s ?$
Commented by FilupSmith last updated on 28/May/16

$note :$ $30, 030 = 2 \times 3 \times 5 \times 7 \times 11 \times 13$
Commented by Rasheed Soomro last updated on 29/May/16

$30, 030 = 2 \times 3 \times 5 \times 7 \times 11 \times 13 (t o t a l f a c t o r s 6)$ $A c t u a l l y w e h a v e t o d i v i d e a b o v e 6 p r i m e s$ $i n t o f o u r g r o u p s w h o s e p r o d u c t s$ $w i l l b e r e q u i r e d f o u r p o s i t i v e n u m b e r s .$ $T w o c a s e s a r e p o s s i b l e .$ $C a s e - I : W h e n g r o u p s c o n t a i n 3, 1, 1, 1 p r i m e s$ $^{6} C_{3} = \frac{6 \times 5 \times 4}{1 \times 2 \times 3} = 20 ways$ $C a s e - I I : W h e n g r o u p s c o n t a i n 2, 2, 1, 1 p r i m e s$ $^{6} C_{2} \times^{4} C_{2} = \frac{6 \times 5}{1 \times 2} \times \frac{4 \times 3}{1 \times 2} = 15 \times 6 = 90 ways$ $Total ways in which the number can be$ $expressed as product of four positive numbers$ $= 20 + 90 = 110$ $R e m e m b e r t h a t o r d e r o f n u m b e r i s n o t i m p o r t a n t .$ $A n d i f o r d e r i s i m p o r t a n t t h e n n u m b e r o f w a y s$ $= 110 \times 4! = 2640$ $I a m n o t c o n f i d e n t .$ $p r a k a s h j a i n, Y o z z i a n d o t h e r s c a n h e l p m e .$
Commented by FilupSmith last updated on 28/May/16

$One thing to consider is that$ $30030 = 2 \times 3 \times 5 \times 7 \times 11 \times 13$ $= 6 \times 5 \times 7 \times 11 \times 13$ $= 2 \times 15 \times 7 \times 11 \times 13$ $etc .$ $= 30 \times 7 \times 11 \times 13$ $= 2 \times 35 \times 11 \times 13$ $etc .$ $i . e . k, l, m, n dont have to be prime$ $the trick, i think, is in the combinations$ $of products, and find the combination$ $out of each one .$
Commented by Rasheed Soomro last updated on 29/May/16

$A s t h e o r d e r {d o e s n}^{'} t m a t t e r i n m u l t i p l i c a t i o n$ $s o I c o n s i d e r e d 6 \times 5 \times 77 \times 13 a n d 13 \times 5 \times 6 \times 77$ $s a m e . A n d t h i s i s w h a t I h a v e u n d e r s t o o d f r o m$ $t h e q u e s t i o n . A n y w a y i f w e h a d a n a n s w e r$ $i n w h i c h o r d e r i s n o t c o n s i d e r e d i m p o r t a n t t h e n$ $c h a n g e i t i n a n s w e r i n w h i c h o r d e r i s i m p o r t a n y t$ $i s v e r y s i m p l e!$
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