Divisibility by three and remainders

I'm sure that anyone can recognise that 3 divides into 9 exactly. So, if we divide the digital root by 3 we will get the remainder. If that remainder is zero then the number is divisible by 3, otherwise we will get 1 or 2. Let's try the number 1416. 1+4+1+6=12. 1+2=3. 3/3=1 with no remainder! Again, there's a simplier way of working this out. Obtaining what I call the 3's digital root. In this case you cross out any digit that is 3, 6 or 9. And any combination of digits that sum to 3, 6, 9, 12 etc. Let's try the number 3485452612. Cross off 3, 6. Also 4+8, 5+4, 1+2. You are left with 5+2. 7/3 = 2 with 1 remainder. Simple. Or, if you prefer, cross off 3, 6 or 9. Replace both 4 and 7 with 1's and replace both 5 and 8 with 2's (the remainders if divided by 3). Then cross off any pairs of 1 and 2, then simply add up any 1's or 2's remaining - divide that by 3 and bingo!

Finally, out of interest, there is another (more complex) method of testing for divisibility by 3 - AND for finding the remainder! First add all the ODD positioned digits (let's call it A), then add all the EVEN positioned digits (B). If A-2B is divisible by 3 so is the original number! If it's the remainder you're after, the result must be between -9 and +9 (a single digit), otherwise do it again and again if necessary. Once you have a single digit, divide it by 3 to obtain a remainder. If it's negative, add 3 giving a value of positive 1 or 2. Let's try 82134682. A=2+6+3+2=13, B=8+4+1+8=21, A-2B=13-42= -29, -(9-4)= -5, -5/3 = -1 and -2 remaining, -2+3=1. The number has a remainder of 1 when divided by 3! Let's try another, 213468. A=1+4+8=13, B=2+3+6=11, A-2B= 13-22= -9. This number is exactly divisible by 3. One interesting observation about this method is that it implies that any two digit number where one digit is twice the other digit will ALWAYS be divisible by three. Let's see - 12, 21, 24, 42, 36, 63, 48 and 84 - all divisible by three! It also implies that any three digit number where the digits are the same will also be divisible by three. 111, 222, 333 ... and 999 are all divisible by three!