Find, with justi�cation, the smallest integer whose �rst digit is 1 and which has the property
that if this digit is transferred to the end of the number, then the number is tripled
A method of solution can be found here:
http://people.missouristate.edu/lesreid/Ans03_96.html
It has to be understood that the "1" is moved to the end of the number, not switched with the last digit.
It is a rather large number, as you will see.
The answer given at the site I provided above is 142857. The "switched" number is 428571
To find the smallest integer that meets the given conditions, we can start by considering the possibilities for the last digit of the number. Since tripling the number is equivalent to appending the first digit to the end, the last digit must be divisible by 3.
The smallest digit divisible by 3 is 3 itself. So, the last digit of the number must be 3.
Now let's consider the possible second to last digit. Since tripling the number involves appending the first digit to the end, we need to find a number that, when tripled, ends with 3.
To get a number that ends with 3 when tripled, we need to find a number that is congruent to 1 modulo 3. The smallest such number is 1 itself.
Therefore, the second to last digit must be 1.
Now we have the first two digits as 1 and 1, and the last digit as 3. We need to find the smallest number with these properties.
The smallest positive integer that has 1 as the first digit, 1 as the second digit, and 3 as the last digit is 113.
Hence, the smallest integer satisfying the given conditions is 113.