Unit 3 · Topic 01 · Factors & Multiples
234 chairs, to be arranged in rows of 3 with none left over.
Kabir started dividing. Anaya added 2, 3 and 4 in her head, got 9, and said yes.
She was finished before he had written the first line.
The hall had 234 chairs and Ms. Rao wanted to know whether they could be set out in rows of 3 with none left over.
Kabir set up a long division. 3 into 2 does not go, 3 into 23 goes 7...
Anaya said "yes" almost immediately.
Kabir kept going and got 78 exactly. "You guessed."
"I added the digits," said Anaya. "Two and three and four is nine. Nine is in the three times table, so 234 is too."
"That can't work. What have the digits got to do with it?"
Ms. Rao took this one. She wrote 234 as 200 + 30 + 4. "Now, 100 divided by 3 leaves remainder 1. And 10 divided by 3 also leaves remainder 1."
She wrote it out: two hundreds leave 2 left over, three tens leave 3 left over, and the four ones leave 4. Total left over: 2 + 3 + 4 = 9.
"The leftovers ARE the digits," she said. "So if the digits add to a multiple of 3, all the leftovers cancel out and nothing remains."
Kabir tried it on 1236. One and two and three and six is twelve. Twelve is a multiple of 3, so yes. He checked with division and it was.
"What about 4?" he asked.
"Different reason. 100 divides exactly by 4, so every hundred, thousand and beyond is already accounted for. Only the last two digits can spoil it — so those are all you check."
Then Anaya asked about 6. Ms. Rao said nothing and wrote 18 on the board, then 2 and 3 underneath.
"Eighteen is even, and one plus eight is nine. So it passes both. And eighteen IS divisible by six."
"6 is 2 times 3," said Kabir. "So it has to pass both tests."
"Careful," said Ms. Rao. "That works because 2 and 3 share no factor. Later you will meet people testing 12 with 2 and 6, and it does not work — 18 passes both of those and is not divisible by 12."
A divisibility rule tells you whether one number divides another exactly, without doing the division.
The last-digit rules are the simplest. A number is divisible by 2 if it ends in an even digit, by 5 if it ends in 0 or 5, and by 10 if it ends in 0.
The digit-sum rules cover 3 and 9. Add all the digits; if that total is a multiple of 3 the number is, and the same for 9. This works because every power of ten leaves a remainder of 1 when divided by 3 or 9, so the leftovers are exactly the digits.
The last-two-digits rule covers 4, because 100 divides exactly by 4 and so cannot contribute a remainder. And 6 is tested by checking 2 and 3 together — but note that divisible by 9 always means divisible by 3, while divisible by 3 does NOT mean divisible by 9.
Take your phone number's last six digits and test that number against every rule in this lesson.
For each rule, say out loud which digits you actually had to look at. You should never need all six for 2, 5 or 10.
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