Let us define the positive number n with four digits a,b,c and d such that n=abcd with a,b,c,d∈Z, 1≤a≤9, 0≤b≤9, 0≤c≤9 and 0≤d≤9. Let us then say that a cool number is a four digit number, say n, such that the two digit numbers written as ab and cd are given by ab=r×s and cd=(r−1)×(s+1) for some non−negative integers r and s, r≠s. For example, 8081 has a=8,b=0 and 80=10×8= while c=8,d=1 and 81=9×9=(10−1)(8+1). So, for n=8081, r=10 while s=8. How many n, for 1000≤n≤9999, are cool? For n∈[1000,9999],n∈Z, how many n exist so that ab=r×s and cd=r×s+1? Call such n warm numbers.


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Question Number 4540 by Yozzii last updated on 06/Feb/16

$L e t u s d e f i n e t h e p o s i t i v e n u m b e r n w i t h f o u r$ $d i g i t s a, b, c a n d d s u c h t h a t n = abcd$ $w i t h a, b, c, d \in Z, 1 ⩽ a ⩽ 9, 0 ⩽ b ⩽ 9,$ $0 ⩽ c ⩽ 9 a n d 0 ⩽ d ⩽ 9 . L e t u s t h e n s a y$ $t h a t a c o o l n u m b e r i s a f o u r d i g i t n u m b e r,$ $s a y n, s u c h t h a t t h e t w o d i g i t n u m b e r s w r i t t e n a s$ $ab a n d cd a r e g i v e n b y ab = r \times s a n d$ $cd = (r - 1) \times (s + 1) f o r s o m e n o n - n e g a t i v e i n t e g e r s$ $r a n d s, r \neq s . F o r e x a m p l e, 8081 h a s$ $a = 8, b = 0 a n d 80 = 10 \times 8 = w h i l e$ $c = 8, d = 1 a n d 81 = 9 \times 9 = (10 - 1) (8 + 1) .$ $S o, f o r n = 8081, r = 10 w h i l e s = 8 .$ $H o w m a n y n, f o r 1000 ⩽ n ⩽ 9999, a r e c o o l ?$ $F o r n \in [1000, 9999], n \in Z, h o w m a n y n e x i s t$ $s o t h a t ab = r \times s a n d cd = r \times s + 1 ? C a l l$ $s u c h n w a r m n u m b e r s .$
Commented by Rasheed Soomro last updated on 06/Feb/16

$F o r w a r m n u m b e r ab = r \times s a n d c \overset{?}{b} = r \times s + 1 ?$ $O r ab = r \times s a n d c d = r \times s + 1 ? P l c o n f i r m .$ $. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .$ $H o w m a n y n u m b e r s a r e t h e r e, w h i c h a r e$ $b o t h w a r m a n d c o o l a t t h e s a m e t i m e ?$ $F o r e x a m p l e$ $4849 48 = 8 \times 6 & 49 = (8 - 1) (6 + 1)$ $4849 48 = 8 \times 6 & 49 = 8 \times 6 + 1$
Commented by Yozzii last updated on 06/Feb/16

$S o r r y f o r t h e t y p o! {I t}^{'} s cd = r \times s + 1 .$
	Answered by Rasheed Soomro last updated on 08/Feb/16

	$C o o l N u m b e r s$ ${L e t}^{'} s a t t a c k t h e p r o b l e m f r o m r - s s i d e .$ $W e h a v e t o f i n d n u m b e r o f p a i r s (r, s)$ $w i t h f o l l o w i n g r e s t r i c t i o n s :$ $^{∙} r \times s m a k e s f i r s t t w o d i g i t s f r o m l e f t (ab)$ $a n d a ⩾ 1,$ $S o, 10 ⩽ r \times s ⩽ 99 . . . . . . . . . . . . . . . . . . . . . . . . . . . . (i)$ $^{∙} (r - 1) (s + 1) m a k e s l a s t t w o d i g i t s$ $00 ⩽ (r - 1) (s + 1) ⩽ 99 . . . . . . . . . . . . . . . . . (i i)$ $^{∙} r a n d s a r e n o n - n e g a t i v e i n t e g e r s a n d r \neq s$ $b u t a s r \times s \neq 0 (f r o m (i)), s o r \neq 0 \land s \neq 0,$ $i - e r a n d s a r e p o s i t v e i n t e g e r s .$ $r, s \in Z^{+} w i t h r \neq s . . . . . . . . . . . . . . . . . . . . . . . (i i i)$ $F r o m (i)$ $10 ⩽ r \times s ⩽ 99 \Rightarrow \frac{10}{r} ⩽ s ⩽ \frac{99}{r}$ $B u t a s r, s \in Z^{+}, S o$ $⌈ \frac{10}{r} ⌉ ⩽ s ⩽ ⌊ \frac{99}{r} ⌋ . . . . . . . . . . . . . . (i v)$ $F r o m (i i)$ $00 ⩽ (r - 1) (s + 1) ⩽ 99 \Rightarrow 0 ⩽ s + 1 ⩽ \frac{99}{r - 1}$ $\Rightarrow - 1 ⩽ s ⩽ \frac{99}{r - 1} - 1$ $\Rightarrow - 1 ⩽ s ⩽ \frac{100 - r}{r - 1}$ $B u t s i n c e r, s \in Z^{+}, S o$ $- 1 ⩽ s ⩽ ⌊ \frac{100 - r}{r - 1} ⌋ . . . . . . . . . . . . . (v)$ $F r o m (i v) & (v)$ $M a x [⌈ \frac{10}{r} ⌉, - 1] ⩽ s ⩽ M i n [⌊ \frac{99}{r} ⌋, ⌊ \frac{100 - r}{r - 1} ⌋]$ $S i n c e r > 0 \frac{10}{r} > - 1, M a x [⌈ \frac{10}{r} ⌉, - 1] = ⌈ \frac{10}{r} ⌉$ $⌈ \frac{10}{r} ⌉ ⩽ s ⩽ M i n [⌊ \frac{99}{r} ⌋, ⌊ \frac{100 - r}{r - 1} ⌋] . . . . (v i)$ $s \neq r . . . . . . . . . . . . . . . . . . . . . . . f r o m (i i i) . . . . . (v i i)$ $F r o m (v i) & (v i i)$ $. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .$ $⌈ \frac{10}{r} ⌉ ⩽ s ⩽ M i n [⌊ \frac{99}{r} ⌋, ⌊ \frac{100 - r}{r - 1} ⌋] \land s \neq r . . . (i x)$ $. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .$ $A b o v e c o n d i t i o n d e t e r m i n e s s, i f w e f i x r .$ $F o r r = 4, s i s g i v e n b y$ $⌈ \frac{10}{4} ⌉ ⩽ s ⩽ M i n [⌊ \frac{99}{4} ⌋, ⌊ \frac{100 - 4}{4 - 1} ⌋]$ $⌈ 2.5 ⌉ ⩽ s ⩽ M i n [⌊ 24.75 ⌋, ⌊ \frac{96}{3} ⌋]$ $3 ⩽ s ⩽ M i n [24, 32]$ $3 ⩽ s ⩽ 24 \land s \neq 4$ $s = 3, 5, 6, 7, . . . 24 T o t a l 21 v a l u e s$ $. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .$ $r = 1; s = 10, 11, . . . . . 99 ∣ r = 31; s = 1, 2$ $r = 2; s = 5, 6, . . ., 49 ∣ r = 32; s = 1, 2$ $r = 3; s = 4, 5, . . . 33 ∣ r = 33; s = 1, 2$ $r = 4; s = 3, \overset{\times}{4}, . . . 24 ∣ r = 34; s = 1, 2$ $r = 5; s = 2, 3, 4, \overset{\times}{5}, . . . 19 ∣ r = 35; s = 1$ $r = 6; s = 2, 3, 4, 5, \overset{\times}{6}, . . . 16 ∣ r = 36; s = 1$ $r = 7; s = 2, 3, . . \overset{\times}{7}, 8, . . . 14 ∣ r = 37; s = 1$ $r = 8; s = 2, 3, . . . \overset{\times}{8}, . . . 12 (38, 1), (39, 1) . . . (50, 1) ∣$ $r = 9; s = 2, 3, . . . . \overset{\times}{9}, 10, 11$ $r = 10; s = 1, 2, . . . 9$ $r = 11; s = 1, 2, . . . 8$ $r = 12; s = 1, 2, . . . 8$ $r = 13; s = 1, 2, . . . 7$ $r = 14; s = 1, 2, . . . 6$ $r = 15; s = 1, 2, . ., 6$ $r = 16; s = 1, 2 . . . 5$ $r = 17; s = 1, 2, . . . 5$ $O u t o f s p a c e$
	Commented by Yozzii last updated on 07/Feb/16

	${I s n}^{'} t i t 10 ⩽ r s ⩽ 99 a n d 0 ⩽ (r - 1) (s + 1) ⩽ 99 ?$
	Commented by Yozzii last updated on 07/Feb/16

	$S i n c e ab = r \times s and cd = (r - 1) (s + 1)$ $\Rightarrow cd = r \times s + r - s - 1 = ab + r - s - 1$ $\Rightarrow cd - ab = r - s - 1$ $10 ⩽ ab ⩽ 99 a n d 00 ⩽ cd ⩽ 99$ $m i n (cd - ab) = m i n (cd) - m i n (ab)$ $= 00 - 10$ $= - 10$ $m a x (cd - ab) = m a x (cd) - m i n (ab)$ $= 99 - 10$ $= 89$ $\Rightarrow 00 - 10 ⩽ cd - ab ⩽ 99 - 10$ $- 10 ⩽ r - s - 1 ⩽ 89$ $- 9 ⩽ r - s ⩽ 90$ $s - 9 ⩽ r ⩽ 90 + s$ $F r o m y o u w e h a v e a l s o \frac{10}{s} ⩽ r ⩽ \frac{99}{s} .$ $C o u l d w e p l o t t h e s e i n e q u a l i t i e s$ $t o g e t t h e c o m m o n r e g i o n ?$
	Commented by Yozzii last updated on 07/Feb/16

	Commented by Yozzii last updated on 07/Feb/16

	Commented by Rasheed Soomro last updated on 07/Feb/16

	$T h a n k s t o m e n t i o n m i s t a k e, I h a v e c o r r e c t e d i t .$
	Commented by Rasheed Soomro last updated on 07/Feb/16

	$NICE APPROACH!$
	Commented by Rasheed Soomro last updated on 07/Feb/16

	$P l d o i t m a n u a l l y a l s o . I m e a n w i t h o u t$ $t h e h e l p o f W o l f r a m A l p h a .$
	Commented by Yozzii last updated on 07/Feb/16

	$T h e b e s t t h i n g I k n o w t o d o$ $i s d r a w t h e g r a p h s m a n u a l l y a n d c o u n t$ $t h e v a l i d i n t e g e r p a i r s . . .$ $I l a c k e x p e r i e n c e i n s o l v i n g s u c h$ $i n e q u a l i t i e s v i a a s h o r t e r m e t h o d .$
	Commented by Rasheed Soomro last updated on 08/Feb/16

	$T h a n k s f o r r e p l y .$
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