Find the number of integer solutions for a×b×c×d=18900 with a,b,c,d≥1.


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Question Number 57985 by mr W last updated on 15/Apr/19

$F i n d t h e n u m b e r o f i n t e g e r s o l u t i o n s$ $f o r a \times b \times c \times d = 18900$ $w i t h a, b, c, d ⩾ 1 .$
	Answered by tanmay.chaudhury50@gmail.com last updated on 15/Apr/19

	$18900 = 3 \times 3 \times 3 \times 7 \times 2 \times 2 \times 5 \times 5$ $18900 = 2^{2} \times 3^{3} \times 5^{2} \times 7^{1}$ $n u m b e r o f d i v i s o r o f 18900 i n c l u d i n g 1 a n d 18900$ $a r e (2 + 1) (3 + 1) (2 + 1) (1 + 1) = 72$ $n o w e x c l u d i n g 1 a n d 18900 \to 72 - 2 = 70$ $f o r e x a m p l e . . .$ $a = 2^{2} = 4$ $b = 3^{3} = 27$ $c = 5^{2} = 25$ $d = 7^{1} = 7$ $4 \times 27 \times 25 \times 7 = 18900$ $n o w i f a = 2$ $b = 2 \times 3^{3} = 54$ $c = 5^{2} = 25$ $d = 7$ $2 \times 54 \times 25 \times 7 = 18900$ $i n t h i s w a y t o t a l i n t e g e r s o l u t i o n$ $a r e = 70$ $s i r p l s c h e c k a m i c o r r e c t . . .$
	Commented by mr W last updated on 15/Apr/19

	$i {d o n}^{'} t h a v e a n a n s w e r y e t s i r . b u t$ $i t h i n k t h e r e m u s t b e m u c h m o r e$ $s o l u t i o n s . e . g .$ $1 \times 1 \times 1 \times 18900$ $1 \times 1 \times 18900 \times 1$ $1 \times 18900 \times 1 \times 1$ $18900 \times 1 \times 1 \times 1$ $a r e 4 d i f f e r e n t v a l i d s o l u t i o n s .$
	Commented by tanmay.chaudhury50@gmail.com last updated on 15/Apr/19

	$y e s s i r . . . i h a v e e x c l u d e d 1 i n m y s o l u t i o n . . .$
	Answered by MJS last updated on 17/Apr/19

	$there are 382 solutions with a ⩽ b ⩽ c ⩽ d but$ $without this restriction there are exactly$ $8000 (I hope I {didn}^{'} t count wrong . . .)$
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