Question

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

Answer 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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