# Number Theory

To borrow from Programming Challenges, number theory in programming contests includes the following:

- Prime Numbers
- Divisibility
- Modular Arithmetic

References

### Primes

In general, I believe 99% of dealing with primes in a problem boils down to loading a list with the first N prime numbers via the Sieve of Erasthones:

primeCnt = 0; for (i=0; i< UPPERLIMIT; i++) a[i] = i; for (i=2; i<UPPERLIMIT; i++) { /* for */ if (0 != a[i]) { /* found a prime */ p[primeCnt] = i; primeCnt++; for (j=i*2; j < UPPERLIMIT; j=j+i) a[j] = 0; } /* found a prime */ } /* for */

Of course, what do you do with the list? Using it to determine prime factors might be common.

### Divisibility

In one sense divisibility is easy -- use mod. Here are some neat rules for divisibility:

- 2 - even, number is divisible if last digit is
- 3 - if repeated sum of digits is 3, 6, or 9 then number is divisible by 3
- 4 - if last 2 digits are divisible by 4
- 5 - ends in 0 or 5
- 6 - both rules 2 and 3 are true
- 7 - repeat: double last digit and subtract from number with last digit removed (integer divided by 10)
458409 45840 - 18 == 45822 4582 - 4 == 4578 457 - 16 == 441 44 - 2 == 42 4 - 4 == 0

- 8 - last 3 digits are divisble by 8
- 9 - sum of digits is 9
- 10 - last digit is a 0
- 11 - alternate subtracting and adding digits
121: 1 - 2 + 1 == 0, so yes 913: 9 - 1 + 3 == 11, so yes 3729: 3 - 7 + 2 - 9 == -11, so yes 14641: 1 - 4 + 6 - 4 + 1 == 0, so yes 12345678906: 1 - 2 + 3 - 4 + 5 - 6 + 7 - 8 + 9 - 0 + 6 == 11, so yes 12345678905: 1 - 2 + 3 - 4 + 5 - 6 + 7 - 8 + 9 - 0 + 5 == 10, so no

- 12 - apply both rules for 3 and 4
- 13 - repeat: multiple last digit by4 and add back to number with last digit removed (integer divided by 10)
2453674 245367 + 16 == 245383 24538 + 12 == 24550 2455 + 0 == 2455 245 + 20 == 265 26 + 20 == 46 4 + 24 == 28 (34 19 37 31 7) 2453672 245367 + 8 == 245375 24537 + 20 == 24557 2455 + 28 == 2483 248 + 12 == 260 26 + 0 == 26

- 14 - even and rule for 7
- 15 - rule for 3 and rule for 5
- 16 - last 4 digits are divisibile by 16

### Modular Arithmetic

Modular arithmetic is sometimes called clock arithmetic. It is the idea that the set of numbers is finite and that you wrap to the begining when you go past the end (like a clock does). Lots of problems rely on these ideas (such as the "every nth person out" problems).

A typical code snippet might be something like:

n = (n + 1) mod MAX

When you think about it, this is how all arithmetic works. `9 + 1 = 10` When you add one to the largest
base 10 digit (9) you get zero for an answer with a carry into the next column. You do the same in modular
arithmetic except for forget the carry.