Find-the-number-of-4-digit-numbers-so-that-when-decomposed-into-prime-factors-have-the-sum-of-prime-factors-equal-to-the-sum-of-the-exponents-

Question Number 211250 by RojaTaniya last updated on 01/Sep/24

F i n d t h e n u m b e r o f 4 d i g i t n u m b e r s

s o t h a t w h e n d e c o m p o s e d i n t o p r i m e

f a c t o r s, h a v e t h e s u m o f p r i m e f a c t o r s

e q u a l t o t h e s u m o f t h e e x p o n e n t s ?

Answered by mr W last updated on 03/Sep/24

s a y t h e n u m b e r i s N = p^{a} q^{b} r^{c} \dots

1000 ⩽ N ⩽ 9999

p, q, r, \dots \in P a n d p < q < r < \dots

a, b, c, \dots \in N

p + q + r + \dots = a + b + c + \dots

{(p^{a} q^{b} r^{c})}_{m i n} = p^{p + q + r - 2} q^{1} r^{1}

{(p^{a} q^{b} r^{c})}_{m a x} = p^{1} q^{1} r^{p + q + r - 2}

n o s o l u t i o n i f

p^{p + q + r - 2} q^{1} r^{1} > 9999 o r

p^{1} q^{1} r^{p + q + r - 2} < 1000

\underset{―}{c a s e 1 : N = p^{a}}

o n l y o n e s o l u t i o n : p = a = 5

s i n c e 3^{3} = 27 < 1000, 5^{5} = 3125, 7^{7} = 823543 > 9999

\underset{―}{c a s e 2 : N = p^{a} q^{b}}

2^{1} 3^{4} = 162 < 1000 \Rightarrow n o s o l u t i o n

2^{6} 5^{1} = 320

2^{3} 5^{4} = 5000, 2^{4} 5^{3} = 2000

2^{8} 7^{1} = 1792

2^{7} 7^{2} = 6272

2^{12} 11^{1} = 45056 > 9999 \Rightarrow n o s o l u t i o n

3^{7} 5^{1} = 10935 > 9999 \Rightarrow n o s o l u t i o n

\underset{―}{c a s e 3 : N = p^{a} q^{b} r^{c}}

2^{8} 3^{1} 5^{1} = 3840

2^{7} 3^{2} 5^{1} = 5760, 2^{7} 3^{1} 5^{2} = 9600, 2^{6} 3^{3} 5^{1} = 8640

2^{10} 3^{1} 7^{1} = 21504 > 9999 \Rightarrow n o s o l u t i o n

\underset{―}{c a s e 4 : N = p^{a} q^{b} r^{c} s^{d}}

2^{14} 3^{1} 5^{1} 7^{1} = 1720320 > 9999 \Rightarrow n o s o l u t i o

s u m m a r y :

t h e r e a r e 9 s u c h 4 - d i g i t n u m b e r s :

1792, 2000, 3125, 3840, 5000,

5760, 6272, 8640, 9600

Commented by Rasheed.Sindhi last updated on 03/Sep/24

E l e g a n t!

Commented by mr W last updated on 03/Sep/24

t h a n k s!

“ b r u t e {f o r c e}^{″} a p p r o a c h, n o t s m a r t,

b u t e f f e c t i v e .

Commented by Rasheed.Sindhi last updated on 03/Sep/24

e^{x} c e l l e n t s i r!