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Olympiad workbook
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IOQM
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Basics of Mathematics

Olympiad workbook · IOQM · Basics of Mathematics

Consider all 6-digit numbers of the form abccba where b is odd....

Consider all 6-digit numbers of the form abccba where b is odd. Determine the number of all such 6-digit numbers that are divisible by 7 .

Correct answer

70

Step-by-step solution

abccba ( b is odd) =a(105+1)+b(104+10)+c(103+102)=a(1001−1)100+a+10b(1001)+(100)(11)c=(7.11.13.100)a−99a+10b(7.11.13)+(98+2)(11)c=7p+(c−a) where p is an integer \begin{aligned} & =a\left(10^5+1\right)+b\left(10^4+10\right)+c\left(10^3+10^2\right) \\ & =a(1001-1) 100+a+10 b(1001)+(100)(11) c \\ & =(7.11 .13 .100) a-99 a+10 b(7.11 .13)+(98+2)(11) c \\ & =7 p+(c-a) \text { where } p \text { is an integer } \end{aligned} Now if c−a\mathrm{c}-\mathrm{a} is a multiple of 7 c−a=7,0,−7c-a=7,0,-7 Hence number of ordered pairs of (a, c) is 14 since b is odd, Number of such number =14×5=70=14 \times 5=70
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