Divisibility by 6 and other numbers, checked through their factors

0:00You have tests for two, for three, for four, for five, for eight, for nine, for ten and for eleven. You do not have one for six. And you are not going to get one, because six does not need one. Six is two times three. So ask two questions you can already answer. Is it even? Do its digits add to a multiple of three? If both answers are yes, the number is a multiple of six.

0:29That is the claim. Now let us find out whether it is true, and then whether the same trick works everywhere else. Four numbers. Thirty-eight, two hundred and twenty-five, one hundred and eighty-six, and sixty-four. Thirty-eight is even, and its digits add to eleven, which is no multiple of three. So it passes one test and fails the other. Two hundred and twenty-five is odd. Its digits add to nine, so it passes the three test and fails the two test.

1:05Sixty-four is even, and its digits add to ten. Same shape as thirty-eight: one yes, one no. One hundred and eighty-six is even, and its digits add to fifteen. Two yeses. Now divide all four by six and check. Only one hundred and eighty-six comes out exact. The two tests got it right. But notice what that set of four does not contain. One number fails only the three test. Two fail only the two test. One passes both.

1:41Nothing there fails both. So let us add one: thirty-five. Thirty-five is odd, and its digits add to eight. Two noes. And thirty-five is not a multiple of six. That is the fourth corner, and it matters, because a rule you have only ever seen agree is not a rule you have tested. So now sweep it. Across nearly three thousand numbers, four hundred and ninety-nine are multiples of six, and the two tests agree with the division on every single number, not just those.

2:18Six is settled. It never needed a rule of its own. So try the same move somewhere harder. Twenty-four. Twenty-four is four times six. You have a test for four and a test for six. Test both, and you have tested twenty-four. That sounds exactly as reasonable as the last one. It is false. Here is the number that kills it. Twelve. Twelve is a multiple of four. Twelve is a multiple of six. Twelve is not a multiple of twenty-four.

2:55One number, and the claim is not weakened but dead. And it is not a freak. Across those same three thousand numbers the four-and-six split gets the answer wrong a hundred and twenty-five times. Now, the tempting reaction is to distrust the tests. Do not. Both tests are perfectly good. The test for four really does find every multiple of four. Seven hundred and forty-nine of them in that range, exactly right.

3:28The test for six really does find every multiple of six. Four hundred and ninety-nine, exactly right. Neither tool is broken. The plan is broken. That distinction is worth keeping. A wrong answer from two correct steps means the way they were combined is where to look. To see where, stop thinking about factors and start counting primes. Every number is built from primes, and what matters is not just which primes but how many copies of each.

4:04Twenty-four is two times two times two times three. Three twos and a three. Read that as an order. To be a multiple of twenty-four, a number must supply three twos and a three. Four is two twos. Six is one two and one three. Eight is three twos. Three is just a three. Now every test becomes a demand for parts, and the question is whether two demands together add up to the order.

4:36Put four and six side by side. Four demands two twos. Six demands one two and one three. Between them, how many twos are guaranteed? Not three. Two. Because the number only has to supply its twos once. Six's two can be one of the two that four already asked for. The two demands overlap, and the overlap is a prime they both need. So the pair guarantees two twos and a three, and twenty-four wants three twos and a three. It is one two short.

5:11Now try three and eight. Three demands a three. Eight demands three twos. No overlap. Three twos and a three, which is exactly the order. That is why the repair works. There is something satisfying hiding here. Four and six do not settle nothing. They settle something exact. Two twos and a three is twelve. So passing both tests guarantees a multiple of twelve. And that is exactly what happens. Two hundred and forty-nine multiples of twelve in that range, and the split finds precisely those.

5:52Twelve is the least common multiple of four and six. Two divisors always force divisibility by their least common multiple, and never by anything larger. That is the whole story. The split told you the truth. You asked it the wrong question. So here is the condition, and it is short. Two tests settle their product exactly when the two share no prime. Share none, and the demands stack up with nothing wasted, so the least common multiple is the product.

6:28Share one, and that prime gets asked for twice and supplied once, so the least common multiple falls short of the product. And this is not a rare trap. Sweep every pair of divisors from two to twenty-five: three hundred pairs, and a hundred and twenty-five of them share a prime. About two in every five. If you pick a split without checking, you are wrong that often. So take the composite divisors you actually run into, and split them properly.

7:03Twelve is four and three. Fifteen is three and five. Eighteen is two and nine. Thirty-six is four and nine. Forty-four is four and eleven. Twenty-four, of course, is three and eight. Look at those pairs. Not one of them shares a prime, and every one of them multiplies to its divisor. Both conditions matter. Two and twelve multiply to twenty-four and share a two, and twelve breaks that one as well.

7:34Six and six multiply to thirty-six and share both primes. Six breaks it immediately. So check two things every time. Do they multiply to the divisor, and do they share nothing? Now a job that looks like ninety pieces of work. Ten numbers, nine divisors, and a yes or a no in every cell. The nine divisors are two, three, four, five, six, eight, nine, ten and eleven. You do not test nine of them. You test seven.

8:09Six is two and three, which are both already in the list. Ten is two and five, also both in the list. Neither shares a prime. So once the two column, the three column and the five column are filled, the six column and the ten column can simply be read off. Two whole columns, twenty cells, done without a single division. And they come out right on all ten numbers, because those splits meet the condition.

8:41One last thing, and it is a habit rather than a rule. Here is a filled-in row for a hundred and twenty-eight. Yes under two. No under four. Yes under eight. You do not need to divide anything to know that row is wrong. Four goes into eight. So anything eight divides, four divides too, always, with no exceptions anywhere. A yes under eight and a no under four is not an unusual number. It is impossible.

9:13And it really is one-way. Plenty of numbers are multiples of four and not of eight: three hundred and seventy-five of them in that range. That row is perfectly ordinary. So a hundred and twenty-eight is a multiple of four, and it is four times thirty-two. There are shortcuts like this for far more divisors than the ones you have met, and the rest need ideas that come later. But the thinking does not change. Count the primes, and check that nothing is being asked for twice.