The digits of a three−digit number A are written in the reverse order to form another three−digit number B. If B>A and B−A is perfectly divisible by 7. Find the range of values of A.


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Question Number 113003 by Aina Samuel Temidayo last updated on 10/Sep/20

$The digits of a three - digit number A$ $are written in the reverse order to$ $form another three - digit number B .$ $If B > A and B - A is perfectly$ $divisible by 7 . Find the range of$ $values of A .$
Commented by Rasheed.Sindhi last updated on 10/Sep/20

$\overset{?}{m i s s} / \overset{?}{m r} A i n a p l r e a d m y c o m m e n t$ $You can't use 'macro parameter character #' in math mode$
Commented by Aina Samuel Temidayo last updated on 10/Sep/20

$Replied ✓$
	Answered by Rasheed.Sindhi last updated on 14/Sep/20

	$A = 100 a + 10 b + c$ $B = 100 c + 10 b + a$ $G i v e n t h a t : (1) B > A \Rightarrow c > a$ $(2) 7 ∣ B - A$ $100 (c - a) + a - c$ $100 (c - a) - (c - a)$ $99 (c - a)$ $7 ∣ 99 (c - a)$ $∵ 7 \in P \land 7 ∤ 99$ $∴ 7 ∣ c - a$ $∴ c - a = 7 k$ $c = a + 7 k$ $A s c > a s o k \neq 0$ $A n d f o r a, c a r e d e c i m a l d i g i t s$ $o f a 3 - d i g i t n u m b e r$ $s o 1 ⩽ a, c ⩽ 9$ $∴ a = 1, 2 & k = 1 \Rightarrow c = 8, 9$ $(a, c) = (1, 8) \land (a, c) = (2, 9)$ $A = 100 + 10 b + 8, 200 + 10 b + 9$ $\begin{matrix} A = 108, 118, 128, . . ., 198 \\ AND \\ A = 209, 219, 229, . . ., 299 \end{matrix}} Ans$
	Commented by Aina Samuel Temidayo last updated on 11/Sep/20

	$Thanks .$
	Commented by Rasheed.Sindhi last updated on 12/Sep/20

	$T h e l a s t p a r t o f m y a n s w e r i s$ $e r r o n e o u s . I^{'} l l c o r r e c t i t .$
	Commented by Rasheed.Sindhi last updated on 14/Sep/20

	$C o r r e c t e d .$
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