The-digits-of-a-three-digit-number-A-are-written-in-the-reverse-order-to-form-another-three-digit-number-B-If-B-gt-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

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

Commented by Rasheed.Sindhi last updated on 10/Sep/20

$$\stackrel{?}{miss}/\stackrel{?}{mr}Ainaplreadmycomment$$$$\text{You can't use 'macro parameter character \#' in math mode}$$

Commented by Aina Samuel Temidayo last updated on 10/Sep/20

$$\mathrm{Replied}✓$$

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

$$A=100a+10b+c$$$$B=100c+10b+a$$$$Giventhat:\left(1\right)B>A\Rightarrow c>a$$$$\left(2\right)7\mid B-A$$$$100(c-a)+a-c$$$$100(c-a)-(c-a)$$$$99(c-a)$$$$7\mid 99(c-a)$$$$\because 7\in \mathbb{P}\wedge 7\nmid 99$$$$\therefore 7\mid c-a$$$$\therefore c-a=7k$$$$c=a+7k$$$$Asc>asok\ne 0$$$$Andfora,caredecimaldigits$$$$ofa3-digitnumber$$$$so1⩽a,c⩽9$$$$\therefore a=1,2\mathrm{\&}k=1\Rightarrow c=8,9$$$$(a,c)=(1,8)\wedge (a,c)=(2,9)$$$$A=100+10b+8,200+10b+9$$$$$$$$\begin{array}{c}A=108,118,128,\dots ,198\\ \mathrm{AND}\\ A=209,219,229,\dots ,299\end{array}\}\mathrm{Ans}$$

Commented by Aina Samuel Temidayo last updated on 11/Sep/20

$$\mathrm{Thanks}.$$

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

$$\mathcal{T}helastpartofmyansweris$$$$erroneous.I^{\prime }llcorrectit.$$

Commented by Rasheed.Sindhi last updated on 14/Sep/20

$$Corrected.$$

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