Attach a 3-digit number to itself (e.g. 547 -> 547547). Divide it by 7.
The remainder will be zero!
Example: Take 826. Then 826826 / 7 = 118118
Now, divide the quotient by 11. The remainder will be zero!
Example: 826826 / 7 = 118118 --> 118118 / 11 = 10738
Again, divide the recent quotient by 13. The remainder will be zero!
Example: 826826 / 7 = 118118 --> 118118 / 11 = 10738 --> 10738 / 13 = 826
Where 826 is the selected number!
Another example:
Take 743!
Concatenate it with itself: 743743
Divide it by 7: 743743 / 7 = 106249
Divide the quotient by 11: 106249 / 11 = 9659
Divide the recent quotient by 13: 9659 / 13 = 743
Why?
7 * 11 * 13 = 1001
Multiplying 1001 by any 3-digit number will be as concatenating it with itself:
100 * 1001 = 100100
945 * 1001 = 945945
So, any number that is obtained by concatenating a 3-digit number with itself can be divided by 1001 and as 1001 is equal to 7 * 11 * 13, it can be divided by 7, 11, and 13.
The remainder will be zero!
Example: Take 826. Then 826826 / 7 = 118118
Now, divide the quotient by 11. The remainder will be zero!
Example: 826826 / 7 = 118118 --> 118118 / 11 = 10738
Again, divide the recent quotient by 13. The remainder will be zero!
Example: 826826 / 7 = 118118 --> 118118 / 11 = 10738 --> 10738 / 13 = 826
Where 826 is the selected number!
Another example:
Take 743!
Concatenate it with itself: 743743
Divide it by 7: 743743 / 7 = 106249
Divide the quotient by 11: 106249 / 11 = 9659
Divide the recent quotient by 13: 9659 / 13 = 743
Why?
7 * 11 * 13 = 1001
Multiplying 1001 by any 3-digit number will be as concatenating it with itself:
100 * 1001 = 100100
945 * 1001 = 945945
So, any number that is obtained by concatenating a 3-digit number with itself can be divided by 1001 and as 1001 is equal to 7 * 11 * 13, it can be divided by 7, 11, and 13.
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