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Question Number 198684 by cortano12 last updated on 23/Oct/23

$ζ$
	Answered by Red1ight last updated on 23/Oct/23

	$24, 88$ $A = 24 \times 88 = 2112$ $P = 24 + 88 = 112$ $2112 \Rightarrow 112 = P$
	Commented by mr W last updated on 23/Oct/23

	$h o w d i d y o u g e t t h e s e n u m b e r s ?$
	Commented by Red1ight last updated on 23/Oct/23

	$by trial and error$
	Commented by mr W last updated on 23/Oct/23

	$t h a n k s!$
	Answered by Rasheed.Sindhi last updated on 23/Oct/23

	$L e t a a n d b a r e 2 - d i g i t n u m b e r s$ $s u c h t h a t :$ $a + b + 2000 = a b$ $a b - a - b = 2000$ $a (b - 1) - b + 1 = 2000 + 1$ $(b - 1) (a - 1) = 2001$ $(b - 1) (a - 1) = 3 \times 23 \times 29$ $(b - 1) (a - 1) = 23 \times 87 = 69 \times 29$ $b - 1 = 23 \land a - 1 = 87$ $b = 24 \land a = 88 ✓$ $b - 1 = 29 \land a - 1 = 69$ $b = 30 \land a = 70 ✓$
	Commented by mr W last updated on 23/Oct/23
	��
	Commented by Rasheed.Sindhi last updated on 23/Oct/23
	��
	Commented by cortano12 last updated on 23/Oct/23

	$nice$
	Answered by mr W last updated on 23/Oct/23

	$s a y t h e t w o 2 - d i g i t n u m b e r s a r e x, y .$ $x y = \underset{―}{2 p q r}$ $x + y = \underset{―}{p q r}$ $x y - (x + y) = 2000$ $x = \frac{2000 + y}{y - 1} ⩽ 99 \Rightarrow y ⩾ 22, s i m i l a r l y x ⩾ 22$ $x = \frac{2001}{y - 1} + 1 = \frac{3 \times 23 \times 29}{y - 1} + 1$ $y - 1 = 23 \Rightarrow y = 24, x = 88$ $y - 1 = 29 \Rightarrow y = 30, x = 70$ $(y - 1 = 3 \times 23 = 69 \Rightarrow y = 70, x = 30)$ $(y - 1 = 3 \times 29 = 87 \Rightarrow y = 88, x = 24)$ $s o t h e t w o n u m b e r s a r e$ $24, 88 o r 30, 70 .$
	Commented by cortano12 last updated on 23/Oct/23

	$great$
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