Common multiples: why the first "idli-vada" is 15

0:00Here's a counting game. A group of children sit in a circle, counting one number each. One, two, three, and on it goes. But there are two special rules. If your number is a multiple of 3, you don't say the number. You say, apple! If your number is a multiple of 5, you say, banana! And if your number is a multiple of 3, and also a multiple of 5, you must say both words together. Apple-banana!

0:32Make a mistake, and you're out of the game. So, pause for a second. Which number gets the very first apple-banana? By the end of this video, you'll see exactly why that number. And you'll meet a puzzle that unlocks this whole chapter. Let's build the two lists, side by side. The apple numbers are the multiples of 3. Three, six, nine, twelve, fifteen, eighteen, twenty-one, and on it goes, for ever.

1:04The banana numbers are the multiples of 5. Five, ten, fifteen, twenty, twenty-five. Two different rhythms, ticking along the same number line. The threes take quick little steps. The fives take bigger, slower ones. And notice, some numbers belong to neither list. Seven is not a multiple of 3, and not a multiple of 5. On turn seven, you just say, seven. Now watch both rhythms at once. Three. Five. Six. Nine. Ten. Twelve. And then. Fifteen!

1:42Both rhythms land on the same number, at the same moment. Let's check that properly. Fifteen divided by three is exactly five. No remainder. So fifteen is on the apple list. And fifteen divided by five is exactly three. No remainder again. So fifteen is on the banana list too. Both lists. Both words. Apple-banana! Do the two rhythms ever meet again? Keep both lists running. Thirty. Both again! Forty-five. Both again! Sixty. Both!

2:21And look at the spacing. The meetings are perfectly evenly spaced. Every fifteen steps. Here's why that has to happen. The moment both rhythms strike together, everything starts over. The threes begin a fresh run of threes, and the fives begin a fresh run of fives. It's exactly like starting the whole game again from zero. So the next meeting must come fifteen steps later. And fifteen after that. For ever.

2:53The very first meeting decides all the others. Numbers like fifteen, thirty, and forty-five have a proper name. Common multiples of 3 and 5. Common, because they belong to both lists at once. Here's a picture for it. One circle holds multiples of 3. The other holds multiples of 5. Where the circles overlap, a number must pass both tests. Not just one. Both. Take nine. Multiple of 3? Yes. Multiple of 5? No. So nine stays in the left circle only.

3:32Twenty. Multiple of 5, but not of 3. Right circle only. But fifteen, and thirty? They pass both tests. Straight into the overlap. Be careful with that middle region. It doesn't mean either list. It means both. This exact picture is Figure 5.1 in the NCERT textbook, and I've redrawn it here. But look closely. Something is missing. Six is a multiple of 3, and it's small enough to belong in this picture. Yet the printed figure skips it.

4:09So let's complete the figure ourselves. There. Now every multiple of 3, up to thirty, is in the picture. And one warning. A drawn circle can only ever hold a few numbers. The real lists never end. Keep that in mind whenever you see this diagram. Now, a bigger question. Suppose the game runs from 1, all the way up to 90. How many times will apple be said? Before we count anything, make a rough guess.

4:42You could write out all ninety turns, one by one. Please don't. Every third number is a multiple of 3. So we just divide. Ninety divided by three. Thirty apple turns. Bananas? Ninety divided by five. Eighteen. And apple-banana? That's every multiple of fifteen. Ninety divided by fifteen. Six double calls. One small note. Those six double calls are already counted inside the thirty, and inside the eighteen. And now our counting has no limits. Run the game to nine hundred. Three hundred apples. One hundred and eighty bananas. Sixty doubles. The same three divisions, just bigger numbers.

5:32One more. Where does the tenth double call happen? Easy. Ten times fifteen. Turn one hundred and fifty. Three divisions. Zero listing. Now let's change the pair, and replay the game. Twos and fives. The first double call lands on ten. And two times five is, ten. Threes and sevens. First double call, twenty-one. Three times seven. Twenty-one again! So it looks like we've found a beautiful shortcut. To find the first meeting, just multiply the two numbers.

6:10Let's test it on fours and sixes. Prediction, four times six. The first double call should land on twenty-four. Watch the track carefully. Four. Six. Eight. Twelve. Wait. Twelve is on both lists! Twelve is a multiple of 4, and a multiple of 6. The first meeting came early. Our multiply shortcut just broke. Hold that thought. Something about this pair is different. One more strange round. A group played this game with a new pair of numbers, and something odd happened.

6:50A lone banana was never heard. Not once. Every single banana turn was already an apple-banana. One of their numbers was 4. So what was the other one? Two, three, five, eight, or ten? Think about when a lone banana disappears. It happens when every multiple of the bigger number is already a multiple of the smaller one. Test eight. Its multiples are eight, sixteen, twenty-four. Is each one a multiple of 4? Yes. Yes. Yes. So with 4 and 8, it works.

7:27Test ten. Is ten a multiple of 4? No. So turn ten would be a lone call. Ten fails. Three and five fail the same way. But here's the sneaky one. Two. With 2 and 4, the bigger number is now 4 itself. And every multiple of 4 is automatically a multiple of 2. It works! Two correct answers. 8, and 2. The textbook's answer page lists only 8. You've just found the one it missed.

8:00Here's a puzzle played backwards. Two circles, but the headings have been erased. Only the overlap survives. Twenty-four. Forty-eight. Seventy-two. Can we recover the headings? Well, the overlap holds the common multiples. And the first one is twenty-four. So we need two numbers whose rhythms first meet at twenty-four. Multiples of 8, and multiples of 12? Eight, sixteen, twenty-four. Twelve, twenty-four. Their first meeting is exactly twenty-four. It fits. But 6 and 8 fit too. So do 3 and 8.

8:41Many answers work, and that's fine. What matters is the check, not the guess. A final challenge from the chapter. Find the smallest number that every one of 1 to 10 divides. Except 7. Strike 7 out. Here's the clever way in. Whatever this number is, 8 must divide it. 9 must divide it. And 5 must divide it. Eight times nine times five. Three hundred and sixty. But wait. What about all the others? Two divides 360, because 2 divides 8. Four divides 8 as well. Three divides 9. Six is two times three, and both are in there. And ten is two times five.

9:31Every single one checks out. Could something smaller work? Try cutting it in half. One hundred and eighty. But 8 does not divide 180. The chain breaks. Three hundred and sixty really is the smallest. Now put 7 back in. Nothing in our building blocks contains a 7, so we simply multiply by 7. Two thousand, five hundred and twenty. So. Why is the first apple-banana at fifteen? Because fifteen is the first number the two rhythms can share. There is nothing magical about fifteen itself.

10:13Change the pair, and the first meeting moves. But here is the mystery we've uncovered. For 3 and 5, the first meeting is exactly three times five. Fifteen. For 2 and 5, it's exactly two times five. Ten. But for 4 and 6, it comes early. Twelve, not twenty-four. And that difference is not about how big the numbers are. It's about what the two numbers secretly share. So look at 4 and 6, side by side. What could they have in common, that 3 and 5 do not?

10:51That is exactly where this chapter goes next. Until then, a challenge for the comments. Can you find a pair of numbers whose very first double call lands exactly on one hundred? There's more than one answer.