If you change the prime number 13 to base 7 you get 16. How can 16 be a prime number? Is it that 16 base 7 is a prime number in base 7 math? Can you give an example of how this could be true?
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Can you find a factor of 16-base-7 in base-7?
For possible numbers, we have 1, 2, 3, 4, 5, 6, 10, 11, 12, 13, 14, 15, 16. Obviously 1 and 16 are out, since those are the prime factors.
What is 16 / 2 in base 7? Since pretty much all calculators operate in base-10 by default, we have to do this manually:
16 – 2 is 14.
14 – 2 is 12.
12 – 2 is 10.
10 – 2 is 5.
5 – 2 is 3.
3 – 2 is 1.
1 – 2 is (-1), so 2 cannot be factor of 16.
Since 2 isn’t a factor, neither can 4, 6, or 11, 13, or 15, since 2 is a factor of those numbers (do you know why?).
So we have 3, 5, and 10, 12, and 14 left as possible factors.
16 – 3 is 13.
13 – 3 is 10.
10 – 3 is 4.
4 – 3 is 1. 3 cannot be a factor.
16 – 5 is 11.
11 – 5 is 3. 5 cannot be a factor.
16 – 10 is 6. 10 cannot be a factor.
16 – 12 is 4. 12 cannot be a factor.
16 – 14 is 2. 14 cannot be a factor.
There are no factors of 16-base-7. Ergo, 16-base-7 is a prime number, just as 13-base-10 is.
Let’s look at a non-prime example; 13-base-7. It looks like a prime, but it isn’t:
Let’s try 2 as a factor.
13 – 2 is 11.
11 – 2 is 6.
6 – 2 is 4.
4 – 2 is 2.
2 – 2 is 0. So 2 is a factor of 13-base-7, as well as 5 (the number of times 2 factors into 13-base-7). Of course this works because 13-base-7 is 10-base-10, so 2 times 5 makes 10.
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