Monday, 17 July 2017

Number Theory | If we repeat a three-digit number twice, to form a six-digit number. The result will be divisible by 7, 11 and 13, and dividing by all three will give your original three-digit number.


Proof:
Suppose "abc" is number which is modified to produce new number like,

            New number = abcabc
                   = 100000*a+10000*b+1000*c+100*a+10*b+c = 100100*a + 10010*b+1001*c
                   = 1001 * (100*a+10*b+c) = 7*11*13 * (100*a+10*b+c).


As 7, 11 and 13 are the factor of the number so it will be always divisible by 7, 11 and 13.

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