Numbers that will not make rectangles

Amma has made 12 laddoos for Diwali and wants to pack them in a flat box, in neat rows, with every row the same length. She can do it lots of ways: 1 row of 12, 2 rows of 6, 3 rows of 4, 4 rows of 3, 6 rows of 2 or 12 rows of 1.

The next day she makes 13 laddoos. Try as she might, the only neat box is one long line: 1 row of 13. Two rows? One row would have 7 and the other 6. Three rows? 4, 4 and 5. Every choice leaves a laddoo sticking out.

Why can 12 be split so many ways while 13 refuses? That one question leads to some of the oldest and most important ideas in mathematics: factors, multiples, and the special numbers called primes.

Think about sharing things fairly. You have 12 pencils and want to share them equally with no pencils left over.

• Between 2 friends: 6 each. Works.
• Between 3 friends: 4 each. Works.
• Between 5 friends: 2 each, with 2 left over. Does not work.

When a number can be shared equally into groups with nothing left over, we say it divides exactly. 12 ÷ 3 = 4 exactly, but 12 ÷ 5 = 2 with a remainder of 2.

The numbers that divide 12 exactly are 1, 2, 3, 4, 6, 12. These are called the factors of 12. You can check each one: 12 ÷ 1 = 12, 12 ÷ 2 = 6, 12 ÷ 3 = 4, 12 ÷ 4 = 3, 12 ÷ 6 = 2 and 12 ÷ 12 = 1. No remainders anywhere.

Now turn the idea around. Start with 4 and keep adding 4: 4, 8, 12, 16, 20, 24, 28, … These are the multiples of 4. They are just the 4 times table: 4 × 1, 4 × 2, 4 × 3 and so on.

Multiples never stop. Whatever multiple you reach, you can always add 4 again. The 4 times table has no last number.

Factors are different. A number has only a few factors, and none of them is bigger than the number itself. 12 has exactly six factors and that is the end of the list.

Here is a picture you can carry in your head. Take a number of square tiles and try to make rectangles from all of them, with no gaps and no tiles left over.

With 12 tiles you can make a 1 by 12 strip, a 2 by 6 rectangle or a 3 by 4 rectangle. (A 4 by 3 is the same rectangle turned on its side.) The side lengths of the rectangles are exactly the factor pairs of 12: 1 × 12, 2 × 6, 3 × 4.

With 7 tiles you can only make a 1 by 7 strip. Any other shape leaves a tile hanging off the edge.

Rangoli makers, gardeners planting rows of saplings and teachers setting out chairs for a school assembly all use this idea. The number of rows and the number in each row must be a factor pair of the total.

The numbers that can only make a single strip have a special name. They are called prime numbers.

A prime number has exactly two factors: 1 and the number itself. 7 has factors 1 and 7. 13 has factors 1 and 13. 29 has factors 1 and 29. Nothing else divides them exactly.

The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.

The word prime comes from the Latin primus, meaning first. Prime numbers are "first" in the sense that they are the basic building blocks: every other counting number bigger than 1 can be made by multiplying primes together, as you will see in later layers.

Numbers like 12, 15, 21 and 27 can make rectangles because they have more than two factors. They are called composite numbers.

Composite means "made of several parts". A composite number can be built by multiplying two smaller numbers (both bigger than 1): 12 = 3 × 4, 15 = 3 × 5, 27 = 3 × 9.

Every even number bigger than 2 is composite, because it can always be split into 2 equal rows. For example 14 = 2 × 7 and 100 = 2 × 50.

Look back at the table. The number 1 has only one factor: itself. A prime needs exactly two different factors, and a composite needs more than two. So 1 is neither prime nor composite. It is in a group of its own.

Mathematicians also have a deeper reason for leaving 1 out of the primes. Primes are the building blocks that other numbers are made from, and each number has only one recipe of primes (12 = 2 × 2 × 3, and no other set of primes multiplies to 12). If 1 counted as a prime, you could write 12 = 2 × 2 × 3 × 1 × 1 × 1 and the recipe would stop being unique. You will meet this idea properly in the Deepen layer.

The number 2 is special in another way. It is the only even prime number.

Every even number can be split into 2 equal groups, so 2 is a factor of every even number. For 2 itself that is fine: its factors are just 1 and 2. But any bigger even number, such as 4, 6, 8 or 100, has at least three factors: 1, 2 and itself. So every even number after 2 is composite.

That means that, apart from 2, all primes are odd. But be careful: not all odd numbers are prime. 9, 15, 21, 25 and 27 are odd and composite.

How can you find all the primes up to 50 without testing every number one at a time? More than 2,200 years ago, a Greek scholar named Eratosthenes, the chief librarian at Alexandria in Egypt, described a clever shortcut. It is called the Sieve of Eratosthenes.

A sieve is like the chalni in the kitchen that lets fine atta through and holds back the lumps. Eratosthenes' sieve lets the primes through and catches the composites.

If you sieve all the way to 100, you find 25 prime numbers:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

It is worth learning the primes up to 30 by heart (there are ten of them), and knowing where to find the rest.

Look at the primes up to 100 again. Some of them sit very close together: 11 and 13, 17 and 19, 29 and 31. Each pair differs by just 2. Primes like these are called twin primes.

Up to 100 there are 8 pairs of twin primes: (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73).

(Why not a gap of 1? Out of any two numbers next to each other, one is even, and the only even prime is 2. So the only primes that differ by 1 are 2 and 3.)

Nobody knows whether twin primes go on forever. Mathematicians have been trying to find out for well over a hundred years. It is one of the most famous unsolved questions in mathematics.

There is one more partner idea. Two numbers are co-prime when the only factor they share is 1.

• 8 and 15: factors of 8 are 1, 2, 4, 8. Factors of 15 are 1, 3, 5, 15. The only one in both lists is 1. So 8 and 15 are co-prime, even though neither of them is prime.
• 8 and 12: both lists contain 1, 2 and 4. They share more than 1, so they are not co-prime.

Co-prime is about a pair of numbers getting along with no shared factors. Prime is about one number on its own.

Primes are not only found in maths books. In parts of the United States, some kinds of periodical cicadas (insects a little like large crickets) spend years underground and then come out all at once, in huge numbers. Some kinds appear every 13 years and others every 17 years. Both are prime numbers.

Scientists think this may help the cicadas avoid predators that have shorter life cycles. A predator that booms every 2, 3, 4 or 6 years will only rarely boom in the same year as a cicada with a prime cycle. You will do the arithmetic for this in the Extend layer.