To find all the numbers which gets reverse when multiplied by 4

To find all the numbers which gets reverse when multiplied by 4

2178 is the smallest number which gets reverse when it is multiplied by 4.

Some other numbers are:

2178 × 4 = 8712

21978 × 4 = 87912

219978 × 4 = 879912

2199978 × 4 = 8799912

21999978 × 4 = 87999912

21782178 × 4 = 87128712  

2199….(n-4)times… 78 x4=87 99….(n-4)times…21

But if number of digits(n) is multiple of 4 then 1 more number is possible and which is

21782178…..2178 × 4 = 87128712……8712

Where n is number of digits in the number

Note: There are no such numbers which gets reverse when they are multiplied by 2,3,5,6,7,8,10,11,12..... and so on.

To find all the natural numbers which gets reverse when multiplied by 9

To find all the natural numbers which gets reverse when multiplied by 9

1089 = 33×33

1089 ×9 = 9801 = 99×99

1089 is the smallest natural number which gets reverse when multiplied by 9.

There is interesting pattern for all the numbers.

 

1089 × 9 = 9801

10989 ×9 = 98901

109989 × 9 = 989901

1099989×9 = 9899901

10999989×9 = 98999901 

10891089×9 = 98019801

 ..............  

If number of digits in number is not  multiple of 4 then only one number is possible.

10999…(n-4) times 89 ×9 =98999…..(n-4) times 01

 Where n is number of digits.

If number of digits in the number  is multiple of 4 then 1 more number is possible.

10891089 × 9 = 98019801

10891089…….1089 × 9 = 98019801……….9801