Different approach in solving this problem?

[Overbearingminds]

If k is a positive integer and if the numeral in the decimal system that represents the product

60k contains only the digits 0 and 2, what is the least possible value of 60k ?

 

 

If this question was asked in a multiple choice exam, one would simply take an answer and find if it is a multiple of 60. What if there were no answers given? For instance what if the question asked to find the first 5 multiples of 60k, that which 60k contains only the digits 0 and 2? The possibilities are not endless, but wouldn't it be rather difficult to pinpoint them?

 

Thanks in advance.

[mathematic]

There aren't too many cases to consider. First simplify the problem by looking at 6k rather than 60k. Once you solve it for 6k, just multiply by 10. Between 0 and 199 there are only 3 numbers 2, 20, 22. Between 200 and 1999 there are only 4 numbers 200, 202, 220, 222 (first hit - so the smallest for the original question is 2220). If you want more, just keep on going.

[baric]

If k is a positive integer and if the numeral in the decimal system that represents the product

60k contains only the digits 0 and 2, what is the least possible value of 60k ?

 

The result needs to contain the digit 2 in multiples of 3 and end with the digit 0. Any number fitting this pattern is guaranteed to be divisible by 60.

 

2220 is the smallest value.

 

5 digits: 22200, 22020, 20220

6 digits: 222000, 220200, 220020, 202200, 202020, 200220

 

You see the pattern.

Edited June 23, 2011 by baric