Unit 3 · Topic 02 · Factors & Multiples
Anaya had 12 tiles and made a rectangle three different ways.
Kabir had 13 tiles and could only make one — a single long line.
Thirteen was not being difficult. It simply had nothing else to offer.
Ms. Rao handed out counters and asked everyone to build every rectangle they could from theirs.
Anaya had 12. She made 1 by 12, then 2 by 6, then 3 by 4. Three different rectangles.
Kabir had 13. He made 1 by 13. Then he sat there. He tried 2 rows and had one left over. He tried 3 and had one left over. He tried every number up to 12.
"Mine's broken," he said.
"Write down the rectangles you found," said Ms. Rao. "Both of you."
Anaya's list of factors: 1, 2, 3, 4, 6, 12. Six factors.
Kabir's list: 1 and 13. Two factors.
"Numbers with exactly two factors are called PRIME," said Ms. Rao. "Numbers with more than two are COMPOSITE. Kabir's 13 is not broken. It is prime."
Anaya asked about 1. She built a rectangle from a single counter: 1 by 1. "That's one factor. So is 1 prime?"
"Prime means exactly TWO factors. One has only one. So it is neither prime nor composite — it sits outside both groups."
Then Ms. Rao put a hundred square on the board and had them cross out multiples. Cross out multiples of 2 after 2 itself. Then multiples of 3 after 3. Then 5, then 7.
"Why do we stop at 7?" asked Kabir.
"What is 11 times 11?"
"121."
"Past 100. So if a number under 100 has a factor, one of its factors must be 10 or less. Checking beyond that is wasted effort."
What survived on the board were 25 numbers. Kabir counted them twice.
Later he tried 91 on his own and declared it prime. Ms. Rao asked him which divisors he had tested. He had done 2, 3 and 5.
"Seven," she said. "Seven times thirteen."
Kabir groaned. "I stopped too early."
"Everybody's first mistake is stopping at 5. Test 7 as well, and 11 if the number is big enough to need it."
Count the factors. A number with exactly TWO factors — one and itself — is PRIME. A number with MORE than two factors is COMPOSITE.
The number 1 has only one factor, so it is neither prime nor composite. It sits outside both groups, and calling it prime breaks the definition.
2 is the only even prime. Every other even number has 2 as a third factor, so it is automatically composite.
To test whether a number is prime, you only need to try divisors up to the point where the divisor squared passes the number. For anything under 100 that means testing 2, 3, 5 and 7 — and stopping at 5 is what makes people wrongly call 49 and 91 prime.
Take 24 counters, coins or dried beans and build every rectangle you can. Write down the factor pair beside each one.
Now try 23 with the same counters. Notice you can only manage one rectangle, and say why.
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