Prime numbers · Martinとコード

Numbers that cannot be smashed

Twelve sweets you can split fairly among 2, 3, 4 or 6 children. But 13 sweets? Whatever group you try, something is always left. Thirteen is a prime — a number that cannot be split fairly.

Put exactly: a prime is a number bigger than 1 that is divisible only by one and by itself. It has no other factor.

Numbers that have more factors (like 12 with factors 2, 3, 4, 6) are called composite — they can be “composed” by multiplying smaller numbers: 12 = 3 × 4 = 2 × 6.

And one? That is neither a prime nor a composite number. It is a loner with a special status — remember that, it is a favourite trick question.

Primes are the building bricks of all numbers: every composite number can be broken into a product of primes (12 = 2 × 2 × 3), and in only one way. That is why they are called the atoms of maths.

After-school club, 17 children, and the leader wants teams. “In twos!” — one left over. “In threes!” — two left over. Filip laughs: “You can try until evening. Seventeen is a prime — it cannot be split into equal groups. Only one by one, or all together.” The leader thinks: “And what then?” Filip: “Either someone joins — eighteen goes in twos, threes, sixes and nines. Or one blows the whistle as referee and sixteen play four on four.” A prime in practice: you tell it because someone is always left over.

💡

Learn the first eight primes by heart: 2, 3, 5, 7, 11, 13, 17, 19. They are the most common “bricks” — you will meet them in every breakdown.

How to check whether a number is prime

A method for numbers up to a hundred — you try small factors in order, with the rules from the last lesson:

Step 1: Is it divisible by two? (Even last digit.) If yes and the number is not 2 itself → composite.

Step 2: Is it divisible by three? (Digit sum.) If yes and it is not 3 itself → composite.

Step 3: Is it divisible by five? (End 0 or 5.) If yes and it is not 5 itself → composite.

Step 4: Is it divisible by seven? Here you have no rule — try dividing (or go through the multiples: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98).

When all four tests fail, a number up to 100 is prime. You do not need to try bigger factors for numbers up to a hundred — if the number had a factor bigger than 10, it must also have a partner smaller than 10 (for example 91 = 7 × 13: the smaller of the pair gives it away first).

Watch three tricks: 1 is not a prime (it has only one factor, not two). 2 is a prime — the only even one. And 91 looks like a prime, but 7 × 13 = 91. Do not skip the seven test.

Is 51 a prime?

It looks “odd and quiet” — exactly like a prime. But the tests:

Two: ends with one, odd ✗ (does not divide).

Three: digit sum 5 + 1 = 6. Six can be divided by three → 51 is divisible by three!

51 : 3 = 17, so 51 = 3 × 17. The number is composite.

The lesson: three is a quiet killer of “quiet” numbers. Always do the digit sum, even when the number looks unbreakable.

Is 47 a prime?

Two: ends with seven, odd ✗.

Three: 4 + 7 = 11, eleven does not go by three ✗.

Five: does not end with 0 or 5 ✗.

Seven: multiples of seven around: 42, 49. Forty-seven is not among them ✗.

All tests failed → 47 is a prime. You cannot split 47 children into equal groups — someone is always left.

Break 60 into a product of primes

I divide in order by the smallest primes:

60 : 2 = 30
30 : 2 = 15
15 : 2 does not go → I try 3: 15 : 3 = 5
5 is a prime — end.

Breakdown: 60 = 2 × 2 × 3 × 5.

Check: 2 × 2 = 4, 4 × 3 = 12, 12 × 5 = 60 ✓. This breakdown into bricks will help you right in the next lesson when looking for common factors and multiples.

⚠️

One is not a prime. A prime must have exactly TWO different factors (1 and itself) — one has only one. And the second trick: 2 IS a prime, even though it is even. It is the only even prime in the world.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. スタート with ten. When it goes well, add more.

Check numbers with tests: two, three (digit sum), five, seven. Answer yes or no.

Practise the questions

On paper

Build your own sieve — the most famous trick for primes, over two thousand years old.

Write the numbers 2 to 50 in a table. Circle 2 and cross out every later multiple of two. Circle 3 and cross out its multiples. The same with 5 and 7. What is left uncrossed are primes — list them.
Check with the tests from the lesson: are 39, 43, 57 and 61 primes? By each one write which test decided.
Break into a product of primes: 24, 45 and 100. By each one do a check by multiplying.
Find all the primes between 20 and 40. How many are there?

Now you 💪

I can say the definition of a prime and I know why one does not belong in it.
I can check a number up to 100 with four tests (2, 3, 5, 7).
I can break a number up to 100 into a product of primes by dividing in order.

✅

Done when: You tell a prime from a composite number with divisibility tests and you break every number up to 100 into prime bricks.

What to take from this lesson

A prime has exactly two factors: one and itself. 2, 3, 5, 7, 11, 13, 17, 19…
A composite number can be built from smaller ones: 12 = 2 × 2 × 3.
One is not a prime and not a composite number. Two is the only even prime.
Test up to a hundred: try factors 2, 3, 5, 7. If they fail, it is a prime.
Breaking into primes = taking a number apart into atoms. It will help with HCF and LCM.