Math Series #15

Algebra Level pending

If A B C D × 9 = D C B A ABCD\times9=DCBA , find the value of the 4-digit integer A B C D \overline{ABCD} , ( 1000 A + 100 B + 10 C + D 1000A+100B+10C+D ) and A , B , C , D A, B, C, D are distinct).

(Source: SASMO Primary)


The answer is 1089.

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1 solution

Chris Lewis
Mar 25, 2021

Note the (original) problem didn't require A , B , C , D A,B,C,D to be distinct; however, we don't need this to find a solution.

A B C D \overline{ABCD} can't be more than 1111 1111 , otherwise 9 × A B C D 9\times \overline{ABCD} would have five digits. So A = 1 A=1 .

Since 1000 A B C D 1111 1000\le \overline{ABCD} \le 1111 , we have 9000 9 × A B C D 9999 9000\le 9 \times \overline{ABCD} \le 9999 and D = 9 D=9 .

Now we have 9 × 1 B C 9 = 9 C B 1 9\times \overline{1BC9} = \overline{9CB1} , so 9 × ( 1000 + 100 B + 10 C + 9 ) = 9000 + 100 C + 10 B + 1 9081 + 900 B + 90 C = 9001 + 100 C + 10 B 89 B + 8 = C \begin{aligned} 9\times (1000+100B+10C+9) &=9000+100C+10B+1 \\ 9081+900B+90C &= 9001+100C+10B \\ 89B+8&=C \end{aligned}

From this last line, it's clear we can't have B > 0 B>0 , so B = 0 B=0 , C = 8 C=8 and A B C D = 1089 \overline{ABCD}=\boxed{1089} .

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