Question

What 4 digit number multiplied by 9 is itself backwards?

Answer 1

For 4-digit numbers, let a, b, c, and d be the digits, so

9*(1000*a+100*b+10*c+d) = 1000*d + 100*c + 10*b + a

8999*a + 890*b = 10*c + 991*d

This implies that d = 10 - a, by looking at last digits, which we

substitute, and get

9990*a + 890*b = 10*c + 9910

When we divide by 10, we get

999*a + 89*b = c + 991

This implies that a = 1, so

89*b = c - 8

This implies that b = 0 and c = 8, and there is only one solution

9*1089 = 9801.

Answer 2

Given :   a 4 digit number is multiplied by 9 its digits get reversed.

To Find : the 4 digit number.

Solution:

ABCD  * 9  =  DCBA

A must be  1    as greater than 1 multiplied by 9 will result in 5 digit number

1BCD  * 9  = DCB1

D must be 9  as only 9 * 9  = 81  

=> 1 BC9  * 9  = 9CB1

C * 9  + 8  = 10X + B     X is carry over

B * 9 +  X  = C      and no carry over   as    1* 9 = 9 in last

so B must be 0

=> X = C

C * 9  + 8  = 10C + 0

=> C= 8

1089 * 9 = 9,801

1089 is 4 digit number

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