1. ## Largest possible value?

For any positive integer N, consider the digits which occur either in N or in 7*N. Let m be the smallest digit among those digits. What is the largest possible value of m?

2. Originally Posted by clarebear14
For any positive integer N, consider the digits which occur either in N or in 7*N. Let m be the smallest digit among those digits. What is the largest possible value of m?
You by far have the oddest problems presented, and I need practice because my hw is going to be basically two of these a night

here's the proof:

2*7 = 14 so any number that starts with 2, will have a 1 carried over to the next place. Example: 28394*7 = 198758 <-- 7*N will always start with 1 if N starts with 2

3*7 = 21 so if N starts with 3 then 7*N starts with 2

here's the whole list:

2 --> 1
3 --> 2
4 --> 2
5 --> 3
6 --> 4
7 --> 4
8 --> 5
9 --> 6

Now it's important to note here that there are two exceptions to this rule, when N starts with 1 then it can be anything, because numbers can carry over, but if N starts with 1 then m would be 1 so no possible value of N starts with 1.

The second is that there can be a number which carries over. However, because that number would be the first digit of one one digit number times 7 and the second digit of another one digit number times 7 with possibly a one carried on top of it, the highest the second digit of 7*N could be if it sent a carry-over to the first number and the first digit is at least 7 would be: 0

That sounded really confusing I know.

Anyway, here's the deal. The highest starting number in 7*N is 7, but that would mean that there is a 0 in 7*n so m would equal 0, that means that the starting number is less than 7 so m is less than 7. It so happens that m can equal 6 and here's an example:

N = 97
7*N = 679
m = 6