Common factors: which jump sizes can land on a number

0:00Here's a game about jumping. I hide a treasure on a number. You pick how long your jump is, and then you're stuck with it. You start at zero, and you can only land where your jump takes you. Say the treasure is on 24, and you choose to jump in fours. Four, eight, twelve, sixteen, twenty, twenty-four. You land right on it. You win. But here's the real question, and I want you to guess before we go on.

0:30How many different jump lengths win this treasure? Most people picture a long list. Hold on to your number, and let's find out. Let's watch that winning jump properly. From zero, every hop of four is exactly the same size. The arcs march along the line. Four, eight, twelve, sixteen, twenty, and the last arc drops onto 24. The jump worked for one reason only. Twenty-four divided by four is six, with nothing left over.

1:05Six perfect jumps, no remainder. That's what landing really means. So which other jump lengths land on 24? Jump in twos, and you land on every even number, so of course you hit 24. Jump in threes. Three, six, nine, and on it goes, straight onto 24. Jump in sixes. Six, twelve, eighteen, twenty-four. Four big jumps. Jump in eights. Eight, sixteen, twenty-four. Three jumps. Jump in twelves, and you're there in two.

1:41But try a jump of five. Five, ten, fifteen, twenty, twenty-five. It steps clean over the treasure. And a jump of seven misses too. Twenty-one, then twenty-eight. It jumps right past. Now, two jump lengths almost everybody forgets. A jump of one. It's slow, but it walks over every single number, so it certainly lands on 24. And a jump of 24 itself. One enormous leap, straight from zero to the treasure.

2:16Both of those win. They feel like cheating, but the rule never said the jump had to be interesting. So our full winning list is one, two, three, four, six, eight, twelve, and twenty-four. Eight winners. Did your guess come close? That list has a name you already know. Each of those numbers is a factor of 24. Your book also calls them divisors. Two words, one idea. And here's a way to be sure you've never missed one. Build them in pairs.

2:51One times twenty-four. Two times twelve. Three times eight. Four times six. Four pairs, eight factors, and the pairs close in from both ends until they meet in the middle. Do it this way and a factor cannot hide from you. Now let's make the game harder. Two treasures at once. One sits on 14, the other on 36. You still only get one jump length. Try jumping in sevens. Seven, fourteen. Straight onto the first treasure.

3:26Keep going. Twenty-one, twenty-eight, thirty-five, forty-two. Watch carefully around 36. Thirty-five, then forty-two. It stepped right over. Now, 36 is much bigger than seven, so the jump certainly travels far enough. But landing was never about travelling far enough. It's about dividing exactly. Thirty-six divided by seven is five, remainder one. That leftover one is the whole story. So which jump lengths collect both treasures? The factors of 14 are one, two, seven, and fourteen.

4:08The factors of 36 are one, two, three, four, six, nine, twelve, eighteen, and thirty-six. Lay the two lists on top of each other and look at the overlap. Only one and two sit in both. Those are the common factors. That's it. Between them the two treasures offer eleven different factors, and only two of them work for both. Try the same thing with 15 and 30. The factors of 15 are one, three, five, and fifteen. Every single one of them also divides thirty.

4:44So here the overlap is the whole of the smaller list. One, three, five, and fifteen. Two pairs, two completely different answers. Now hold on. Fourteen and thirty-six aren't small numbers, and they share almost nothing. So does a bigger pair give you a longer list of common factors? No. Size has nothing to do with it. But two things are always true, and they're worth knowing. First, one is a factor of absolutely every number, so every pair shares at least one.

5:21The overlap is never empty. Second, a factor can never be bigger than the number it divides. So the common factors of a pair are trapped between one and the smaller number. The list always ends. Here's the same idea hiding in a grid, and this one is yours to crack. The numbers 31 to 70, ten to a row. Some cells are shaded, some are ringed. The shaded ones are 33, 36, 39, 42, and on in that rhythm.

5:57The ringed ones are 32, 36, 40, 44, and on in a different rhythm. Pause and work out what each rhythm is counting in, before I tell you. The shaded cells are the multiples of 3. The ringed cells are the multiples of 4. And three cells carry both marks. Thirty-six, forty-eight, and sixty. Those are the common multiples of 3 and 4, and they're exactly the multiples of twelve.

6:29Now put the two games side by side, because they behave completely differently. Common multiples of 3 and 4 start at twelve and keep going. Twenty-four, thirty-six, forty-eight, for ever. That list runs off the edge of the screen and never stops. Common factors of a pair sit in a short list that starts at one and stops at the smaller number. It's not a coincidence. It's forced. A multiple has nowhere it has to stop. A factor is fenced in by the number it divides.

7:06Endless one way, finite the other. Let's run the machine backwards and hunt numbers from clues. Find a number under 40 that has 7 as a factor, and whose digits add up to eight. The multiples of seven under forty are seven, fourteen, twenty-one, twenty-eight, thirty-five. Check the digit sums. Only thirty-five gives three plus five, which is eight. That's the number. Try another. Under 100, with both 3 and 5 as factors, and two digits that differ by one.

7:43Having both three and five means it's a multiple of fifteen. Fifteen, thirty, forty-five, sixty, seventy-five, ninety. Which has digits differing by one? Forty-five. Four and five. Now a few quick ones. The common factors of 20 and 28 are one, two, and four. For 35 and 50, just one and five. You can do it for three numbers at once too. Four, eight and twelve all share one, two, and four.

8:17Five, fifteen and twenty-five share one and five. Which jump lengths reach both 28 and 70? Their common factors. One, two, seven, and fourteen. And which multiples of 40 sit between 310 and 410? Three hundred and twenty, three hundred and sixty, four hundred. One more that catches people out. Name three multiples of 25 that are not multiples of 50. Twenty-five, seventy-five, and a hundred and twenty-five. The odd ones in the twenty-five rhythm.

8:57One last idea, and it's a strange one. Take 28 and list its factors. One, two, four, seven, fourteen, twenty-eight. Add them all up. That comes to fifty-six, which is exactly twice twenty-eight. A number whose factors add to twice itself is called a perfect number. There's one hiding between 1 and 10. Its factors are one, two, three, and six, and they add to twelve. Twelve is twice six. So six is perfect.

9:31And that's the heart of both these videos. A jump lands when it divides exactly. Multiples are where you can get to. Factors are what you're built from. Here's the open question. Perfect numbers are extremely rare. Six, then twenty-eight, then four hundred and ninety-six. Nobody has ever found an odd one, and nobody has proved that none exists. That question has been open for over two thousand years. Tell me in the comments if you can find the one after 496.