f-is-a-function-that-assigns-to-each-natural-number-multiplication-of-its-digits-for-exampl-f-200-0-f-128-1-2-8-16-find-the-value-of-f-100-f-101-f-

Question Number 156775 by mnjuly1970 last updated on 15/Oct/21

f i s a f u n c t i o n t h a t a s s i g n s

t o e a c h n a t u r a l n u m b e r

m u l t i p l i c a t i o n o f i t s d i g i t s

f o r e x a m p l : f (200) = 0

f (128) = 1 \times 2 \times 8 = 16

f i n d t h e v a l u e o f :

f (100) + f (101) + f (102) + \dots f (999) = ?

Commented by Rasheed.Sindhi last updated on 15/Oct/21

\underset{a = 1, b = 0, c = 0}{\sum^{a = 9, b = 9, c = 9}} a b c = 91125

Answered by Rasheed.Sindhi last updated on 15/Oct/21

100 ⩽ \overset{―}{a b c} ⩽ 999

c = 0, 1, 2, \dots, 9

\overset{―}{a b 0}, \overset{―}{a b 1}, \overset{―}{a b 2}, \dots \overset{―}{a b 9}

f (\overset{―}{a b 0}) + f (\overset{―}{a b 1}) + f (\overset{―}{a b 2}) +, \dots + f (\overset{―}{a b 9})

= a . b . 0 + a . b . 1 + a . b . 2 + \dots + a . b . 9

= a b (0 + 1 + 2 + 3 + \dots + 9)

= a b (\frac{9 \times 10}{2}) = a b (45)

b = 0, 1, 2, \dots, 9

a . b (45) :

45 a . 0 + 45 a . 1 + 45 a . 2 + \dots + 45 a . 9

= 45 a (0 + 1 + 2 + \dots + 9) = 45^{2} a

a = 1, 2, 3, \dots, 9

45^{2} a :

45^{2} . 1 + 45^{2} . 2 + 45^{2} . 3 + \dots + 45^{2} . 9

45^{2} (1 + 2 + 3 + \dots + 9) = 45^{3} = 91125

B y m y s o n, Arshad Soomro

Commented by mnjuly1970 last updated on 16/Oct/21

v e r y n i c e t h a n k s a l o t \dots

Commented by qaz last updated on 15/Oct/21

S = \underset{a = 1}{\sum^{9}} \underset{b = 1}{\sum^{9}} \underset{c = 1}{\sum^{9}} abc = \underset{a = 1}{\sum^{9}} a \underset{b = 1}{\sum^{9}} b \underset{c = 1}{\sum^{9}} c = {(\frac{9 \cdot 10}{2})}^{3} = 91125

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