To see if a number, say 562437487, is divisible by 3, you just add up the digits of its de...

To see if a number, say 562437487, is divisible by 3, you just add up the digits of its decimal representation, and see if the result is divisible by 3. (5 + 6 + 2 + 4 + 3 + 7 + 4 + 8 + 7 = 46, so it is not divisible by 3.)

To see if the same number is divisible by 11, you can do this: subdivide the number into pairs of digits, from the right-hand end (87, 74,43, 62, 5), add these numbers, and see if the sum is divisible by 11 (if it’s too big, repeat).

How about 37? To see if the number is divisible by 37, subdivide it into triples from the end (487, 437, 562) add these up, and see if the sum is divisible by 37.

This is true for any prime p other than 2 and 5. That is, for any prime p f/=2, 5, there is an integer r such that in order to see if p divides a decimal number n, we break n into r-tuples of decimal digits (starting from the right-hand end), add up these r-tuples, and check if the sum is divisible by p.

(a) What is the smallest such r for p = 13? For p = 17?

(b) Show that r is a divisor of p − 1.