Find all four-digit numbers abba with the property aa . 10b= abba
abba if a

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1. Given :– Find all four-digit numbers abba with the property aa . 10b= abba if a ≠ b . [ Note :- aa is a two digit number and 10b is a three digit number. ]

   Answer :–

   given that, abba is four digit number,

   so,

   → abba = 1000a + 100b + 10b + a

   → abba = 1000a + a + 100b + 10b

   → abba = 1001a + 110b

   now,

   → aa = 2 digit number = 10a + a = 11a .

   → 10b = 3 digit number = (100 + b)

   then,

   → aa . 10b = abba

   → 11a . (100 + b) = 1001a + 110b

   → 11a . (100 + b) = 11(91a + 10b)

   → a . (100 + b) = (91a + 10b)

   → 100a + ab = 91a + 10b

   → 100a – 91a = 10b – ab

   → 9a = b(10 – a)

   putting value of a now,

   • if a = 1 => b = 1 , but a ≠ b .
   • if a = 2 => b ≠ natural number .
   • if a = 3 => b ≠ natural number .
   • if a = 4 => b = 6 .
   • if a = 5 => b = 9
   • if a = 6 ≠ natural number .
   • if a = 7 => b = 21 => two digit number .
   • if a = 8 => b = two digit number .
   • if a = 9 => b = two digit number .

   therefore, required four digits numbers abba are :-

   • 4664
   • 4664 5995 .

   Hence , 4664 and 5995 are four-digit numbers .

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