Emordnilap palindromE

Number Theory Level 3

How many 4-digit palindromes are multiples of 23?


The answer is 3.

Alex Segesta by Alex Segesta , 20, USA

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2 solutions

Siva Prasad
Sep 18, 2015

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Sunil Pradhan
Mar 12, 2015

The numbers are of the type aaaa or abba

i.e. 1000a + 100a + 10a + a = 1001a to be divisible by 23 a must be 23 not possible so there are no such numbers.

other types are 1000a + 100b + 10c + a type = (1001a + 110b) divisible by 23

to make it easier it is written as (989a + 92b) + (12a + 18b)

out of these (989a + 92b) is divisible by 23

so (12a + 18b ) = 4(2a + 3b) must be divisible by 23 to choose a and b Let a = 1, 2, 3, ...

when a = 1, 4 and 7 then b= 7, 5, 9 (2a + 3b) divisible by 23

numbers are 1771, 4554, 7992

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