5.6 Summary
Key Points
- Parity (even/odd) is carried through addition and subtraction predictably: a\pm b always share a parity, so any chain a\pm b\pm c\pm d has one fixed parity. We can judge whether an expression is even without computing it.
- A claim about factors and multiples may be always, sometimes, or never true, algebra and pictures decide, far faster than checking cases.
- If a divides M and N, then a divides M+N and M-N. If A is divisible by k, every multiple of A is divisible by k, and A is divisible by every factor of k. If A is divisible by k and m, it is divisible by \operatorname{LCM}(k,m).
- Divisibility tests: a number is divisible by 9 (or 3) iff its digit sum is; by 11 iff the alternating digit sum is 0 or a multiple of 11; and by 7 iff “the rest minus twice the units digit” is. These all flow from how powers of 10 behave modulo 9, 3, 11 and 7.
- The digital root (repeated digit sum) equals the remainder on division by 9, the basis of “casting out nines”.
- Cryptarithms yield to reasoning about leading digits, parity and place value.