The sum of the first two digits of a three digit number is equal to the third digit. If the digits are reversed, the number so obtained is 198 more than the original number. What are the possible original numbers?

Shohan

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2025-01-04 04:42

1 Answers

Shohan

$L e t, x b e t h e 100 t h d i g i t; y b e t h e 10 t h d i g i t a n d z b e t h e u n i t d i g i t$

$∴ T h e n u m b e r i s = (100 x + 10 y + z) A c c o r d i n g q u e s t i o n t o, (x + y) = z . . . . . . . (i)$

$(100 x + 10 y + z) = x + 10 y + 100 z - 198 = x - 100 x + 10 y - 10 y + 100 z - z = 198 = 99 z - 99 x = 198 = 99 (z - x) = 198 = (z - x) = \frac{198}{99} = 2$

$= x + y - x = 2 [P u t t i n g z = x + y] ∴ y = 2 [H e r e, 10 t h d i g i t a l w a y s w i l l b e 2] W h e n x = 1 t h e n t h e p o s s i b l e n u m b e r i s = 1 + 2 = 3 : h e r e b y t h e n u m b e r i s 123 . ∴ x = 2 t h e n t h e p o s s i b l e n u m b e r i s 2 + 2 = 4; h e r e b y t h e n u m b e r i s 224 . ∴ x = 3 t h e n t h e p o s s i b l e n u m b e r i s = 3 + 2 = 5; h e r e b y t h e n u m b e r i s 325 . ∴ x = 4 t h e n t h e p o s s i b l e n u m b e r i s = 4 + 2 = 6 : h e r e b y t h e n u m b e r i s 426 . ∴ x = 5 t h e n t h e p o s s i b l e n u m b e r i s = 5 + 2 = 7; h e r e b y t h e n u m b e r i s 527 . ∴ x = 6 t h e n t h e p o s s i b l e n u m b e r i s = 6 + 2 = 8; h e r e b y t h e n u m b e r i s 62 . ∴ x = 7 t h e n t h e p o s s i b l e n u m b e r i s = 7 + 2 = 9 : h e r e b y t h e n u m b e r i s 729 . T h e n u m b e r c a n b e 123, 224, 325, 426, 527, 628, 729 .$

8 views Answered 2024-12-14 13:31

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