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Question Number 184477 by Shrinava last updated on 07/Jan/23

Commented by JDamian last updated on 07/Jan/23
I don't even understand your question. Would you mind express in math language?
Commented by mr W last updated on 07/Jan/23

$w h e n C (n) = c o r e o f n u m b e r n, t h e n$ $C (21) = 18$ $C (12) = 9$
	Answered by Rasheed.Sindhi last updated on 07/Jan/23
	$•tu^(−) is 2-digit number.Clearly u∈{0,1,2,...,9} & t∈{1,2,3,...,9} •Core of tu^(−) is either 9 or 18 and this depends upon values of u and t as follows: determinant ((u,(values of t_(for core 9_(/total) ) ),(values of t_(for core 18_(/total) ) )),(0,(1≤t≤9 9),(− 0)),(1,(− 0),(1≤t≤9 9)),(2,(t=1 1),(2≤t≤9 8)),(3,(t=1,2 2),(3≤t≤9 7)),(4,(1≤t≤3 3),(4≤t≤9 6)),(5,(1≤t≤4 4),(5≤t≤9 5)),(6,(1≤t≤5 5),(6≤t≤9 4)),(7,(1≤t≤6 6),(7≤t≤9 3)),(8,(1≤t≤7 7),(t=8,9 2)),(9,(1≤t≤8 8 ),(t=9 1)),(,(TOTAL 45),( 45))) Sum of the cores=9×45+18×45 =1215$
	$∙ \overset{―}{t u} i s 2 - d i g i t n u m b e r . C l e a r l y$ $u \in {0, 1, 2, . . ., 9} & t \in {1, 2, 3, . . ., 9}$ $∙ C o r e o f \overset{―}{t u} i s e i t h e r 9 o r 18$ $a n d t h i s d e p e n d s u p o n v a l u e s o f$ $u a n d t a s f o l l o w s :$ $\begin{array}{cccccccccccc} u & \underset{\underset{/ t o t a l}{f o r c o r e 9}}{v a l u e s o f t} & \underset{\underset{/ t o t a l}{f o r c o r e 18}}{v a l u e s o f t} \\ 0 & 1 ⩽ t ⩽ 9 9 & - 0 \\ 1 & - 0 & 1 ⩽ t ⩽ 9 9 \\ 2 & t = 1 1 & 2 ⩽ t ⩽ 9 8 \\ 3 & t = 1, 2 2 & 3 ⩽ t ⩽ 9 7 \\ 4 & 1 ⩽ t ⩽ 3 3 & 4 ⩽ t ⩽ 9 6 \\ 5 & 1 ⩽ t ⩽ 4 4 & 5 ⩽ t ⩽ 9 5 \\ 6 & 1 ⩽ t ⩽ 5 5 & 6 ⩽ t ⩽ 9 4 \\ 7 & 1 ⩽ t ⩽ 6 6 & 7 ⩽ t ⩽ 9 3 \\ 8 & 1 ⩽ t ⩽ 7 7 & t = 8, 9 2 \\ 9 & 1 ⩽ t ⩽ 8 8 & t = 9 1 \\ T O T A L 45 & 45 \end{array}$ $S u m o f t h e c o r e s = 9 \times 45 + 18 \times 45$ $= 1215$
	Answered by Rasheed.Sindhi last updated on 08/Jan/23
	$AnOther way: •tu^(−) is 2-digit number.Clearly u∈{0,1,2,...,9} & t∈{1,2,3,...,9} Observations: •Number of 2-digit numbers =99−9=90 •Core of tu^(−) is either 9 or 18 •All numbers in which t<u have core 9, other numbers(in which t≥u but u≠0) have core 18 •In case u=0 core of t0^(−) is 9 although t>u • Number of the numbers having core 9 is 45.^★ So sum of cores is 9 × 45 = 405 • Obviously number of the remaining numbers(in which t≥u and having cores 18) is 90−45=45.So the sum of cores 18×45=810 •Sum of cores of all 2-digit number is therefore 405+810=1215 ^★ The way of counting is given in my other answer.The better way of counting may be suggested by sir mr W$
	$AnOther way :$ $∙ \overset{―}{t u} i s 2 - d i g i t n u m b e r . C l e a r l y$ $u \in {0, 1, 2, . . ., 9} & t \in {1, 2, 3, . . ., 9}$ $O b s e r v a t i o n s :$ $∙ N u m b e r o f 2 - d i g i t n u m b e r s$ $= 99 - 9 = 90$ $∙ C o r e o f \overset{―}{t u} i s e i t h e r 9 o r 18$ $∙ A l l n u m b e r s i n w h i c h t < u h a v e$ $c o r e 9, o t h e r n u m b e r s (i n w h i c h$ $t ⩾ u b u t u \neq 0) h a v e c o r e 18$ $∙ I n c a s e u = 0 c o r e o f \overset{―}{t 0} i s 9$ $a l t h o u g h t > u$ $∙ N u m b e r o f t h e n u m b e r s h a v i n g$ $c o r e 9 i s 45 .^{★}$ $S o s u m o f c o r e s i s 9 \times 45 = 405$ $∙ O b v i o u s l y n u m b e r o f t h e r e m a i n i n g$ $n u m b e r s (i n w h i c h t ⩾ u a n d h a v i n g$ $c o r e s 18) i s 90 - 45 = 45 . S o t h e s u m$ $o f c o r e s 18 \times 45 = 810$ $∙ S u m o f c o r e s o f a l l 2 - d i g i t n u m b e r$ $i s t h e r e f o r e 405 + 810 = 1215$ $^{★} T h e w a y o f c o u n t i n g i s g i v e n i n$ $m y o t h e r a n s w e r . T h e b e t t e r w a y$ $o f c o u n t i n g m a y b e s u g g e s t e d$ $b y s i r m r W$
	Commented by mr W last updated on 08/Jan/23

	$i t h i n k y o u h a v e g r e a t l y s o l v e d s i r .$ $n u m b e r s w i t h t < u h a v e t h e c o r e$ $9, w h i l e n u m b e r s w i t h t ⩾ u h a v e$ $t h e c o r e 18^{)} .$ $t h i s i s j u s t a n o b s e r v a t i o n . t o b e$ $m a t h e m a t i c a l l y s t r i c t, i t m u s t b e$ $p r o v e d .$ $^{)} t h e r e a r e e x c e p t i o n s! e . g . t h e c o r e$ $o f 20 i s 9, n o t 18 .$
	Commented by Rasheed.Sindhi last updated on 08/Jan/23

	$G r a t e f u l s i r! I^{'} v e a d d e d n o w t h e$ $e x c e p t i o n . P l r e r e a d m y a n s w e r .$ $A s f a r a s t h e p r o o f i s c o n c e r n e d I$ $k n o w o n l y o n e w a y t h a t i s t o t e s t$ $t h e p r o p o s i t i o n f o r a l l n u m b e r s (10,$ $11, 12, . . . 99) .$ $\begin{array}{cccccccccc} 10 & 20 & 30 & 40 & 50 & 60 & 70 & 80 & 90 \\ 11 & 21 & 31 & 41 & 51 & 61 & 71 & 81 & 91 \\ 12 & 22 & 32 & 42 & 52 & 62 & 72 & 82 & 92 \\ 13 & 23 & 33 & 43 & 53 & 63 & 73 & 83 & 93 \\ 14 & 24 & 34 & 44 & 54 & 64 & 74 & 84 & 94 \\ 15 & 25 & 35 & 45 & 55 & 65 & 75 & 85 & 95 \\ 16 & 26 & 36 & 46 & 56 & 66 & 76 & 86 & 96 \\ 17 & 27 & 37 & 47 & 57 & 67 & 77 & 87 & 97 \\ 18 & 28 & 38 & 48 & 58 & 68 & 78 & 88 & 98 \\ 19 & 29 & 39 & 49 & 59 & 69 & 79 & 89 & 99 \end{array}$ $B l u e n u m b e r s h a v e c o r e 9$ $R e d n u m b e r s h a v e c o r e 18$
	Commented by mr W last updated on 08/Jan/23

	$y o u r s o l u t i o n i s g r e a t!$ $a s f o r t h e p r o o f, s e e m y a t t e m p t .$
	Answered by mr W last updated on 08/Jan/23

	$t h e r e a r e 3 t y p s o f 2 - d i g i t n u m b e r s$ $t y p 1 : l i k e 10, 20$ $9 n u m b e r s, c o r e o f e a c h i s 9$ $t y p 2 : l i k e 11, 22$ $9 n u m b e r s, c o r e o f e a c h i s 18$ $t y p 3 : l i k e 12, 23 o r 21, 32$ $9 \times 8 = 72 s u c h n u m b e r s . o n e h a l f o f t h e m,$ $l i k e 12, h a v e t h e c o r e 9, t h e o t h e r h a l f,$ $l i k e 21, h a v e t h e c o r e 18 .$ $t h e s u m o f t h e c o r e s o f a l l 2 - d i g i t$ $n u m b e r s :$ $9 \times 8 + 9 \times 18 + 36 \times 9 + 36 \times 18 = 1215$
	Commented by mr W last updated on 09/Jan/23

	$\underset{―}{a b o u t t h e c o r e o f 2 - d i g i t n u m b e r s}$ $t h e c o r e o f a 2 - d i g i t n u m b e r i s$ $d e f i n e d a s t h e s u m o f d i g i t s o f t h e$ $9 t i m e s o f t h i s n u m b e r .$ $e x a m p l e n = 25$ $9 n = 9 \times 25 = 225$ $2 + 2 + 5 = 9$ $t h a t m e a n s t h e c o r e o f 25 i s 9 .$ $c a s e 1 : n u m b e r s w i t h d i g i t z e r o$ $n = \underset{―}{a 0}$ $9 n = 10 n - n = a 00 - a 0$ $t h e f i r s t d i g i t o f i s a - 1, t h e$ $s e c o n d d i g i t i s 10 - a, t h e t h i r d i s 0 .$ $s u m o f a l l d i g i t s i s a - 1 + 10 - a + 0 = 9$ $t h a t m e a n s t h e c o r e o f \underset{―}{a 0} i s a l w a y s 9 .$ $c a s e 2 : n u m b e r s w i t h o u t d i g i t z e r o$ $n = \underset{―}{a b}$ $9 n = 10 n - n = a b 0 - a b$ $t h e t h i r d d i g i t i s 10 - b$ $t h e s e c o n d d i g i t i s b - 1 - a i f b - 1 ⩾ a$ $o r 10 + b - 1 - a i f b - 1 < a$ $t h e f i r s t d i g i t i s a i f b - 1 ⩾ a$ $o r a - 1 i f b - 1 < a$ $s u m o f a l l d i g i t s i s$ $10 - b + b - 1 - a + a = 9 i f b - 1 ⩾ a$ $o r 10 - b + 10 + b - 1 - a + a - 1 = 18 i f b - 1 < a$ $t h a t m e a n s$ $t h e c o r e o f \underset{―}{a b} i s {\begin{cases} 9 i f a < b \\ 18 i f a ⩾ b \end{cases}$
	Commented by mr W last updated on 09/Jan/23

	$t h a n k s!$
	Commented by Rasheed.Sindhi last updated on 10/Jan/23

	$X c e l l e n t S i r! T h a n k s a l o t .$
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