Number 3

If a certain 4-digit integer A B C D \overline{ABCD} is multiplied by 4, the answer comes out to be D C B A \overline{DCBA} .Find the value of A B C D \overline{ABCD} .


The answer is 2178.

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1 solution

4 × A B C D = D C B A 4 × \overline { ABCD } = \overline {DCBA}

4 × A B C D 9999 A B C D 2499 4 × \overline {ABCD} ≥ 9999 → \overline {ABCD} ≥ 2499

Then A 2 A ≤ 2

4 D A ( m o d 10 ) . 4D \equiv A\pmod{10}.

A is even A = 2 → A = 2

4 D 2 ( m o d 10 ) 4D \equiv 2\pmod{10} D 3 ( m o d 5 ) D \equiv 3\pmod{5}

D = 3 D = 3 or D = 8 D=8

4 × 2 B C D = D C B A 8000 4 × \overline {2BCD} = \overline {DCBA} ≥ 8000 D = 8 → D = 8

4 ( 2000 + 100 B + 10 C + 8 ) = 8000 + 100 C + 10 B + 2 4 (2000 + 100B + 10 C + 8) = 8000 + 100C + 10 B + 2

8000 + 40 B + 40 C + 32 = 8000 + 100 C + 10 B + 2 8000 +40B + 40C + 32 = 8000 + 100C + 10 B + 2

13 B = 2 C 1 13B = 2C - 1

2 C 11 17 2C -1 1 ≤ 17 13 B 17 → 13B ≤ 17

B = 0 B = 0 or B = 1 B = 1

4 D C B A 4\mid \overline {DCBA} 4 B 2 4\mid \overline {B2} B = 1 → B = 1

13 = 2 C 1 13 = 2C - 1 C = 7 → C = 7

Finally 4 ( 2178 ) = 8712 4 (2178) = 8712

Nice Solution........................

Soumya Shrivastva - 5 years, 1 month ago

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Did the same . Can u generalize it with n number of letters.

Aditya Kumar - 5 years ago

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