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← Level 3 – Fraction kid

12 / 30 ⏱ 15 minutes

Divisibility: 2, 3, 5, 10

How to tell without dividing whether a number can be divided with no remainder.

You can tell without counting

Can the number 738 be split into three equal piles with no remainder? You can try dividing… or you can tell in two seconds by looking. Today you will learn detective rules of divisibility.

“A number is divisible by three” means: when you divide by three, no remainder comes out. 12 is divisible by three (12 : 3 = 4, cleanly), 13 is not (remainder 1).

The magic is that you can tell divisibility from the digits, with no dividing:

What is it for? Simplifying fractions (you look for common factors!), splitting people into groups, and in two lessons primes. Divisibility is an X-ray — you see into a number without taking it apart.

The teacher wants to split 87 cups into threes for experiments. “Does it come out exactly?” she asks the class. Jonáš starts dividing in columns, but Sofie is faster: “Eight plus seven is fifteen. Fifteen can be divided by three — so 87 can too!” The teacher nods: “Digit sum. Right.” Jonáš meanwhile finishes: 87 : 3 = 29, no remainder. The same result, but Sofie had it in two seconds. “How did you know?” “Three you tell from the sum of the digits. Two and five from the last digit. I’ll teach you at break.”

💡

With big numbers you can add the digit sum once more: 6987 → 6+9+8+7 = 30 → 3+0 = 3. Three divides it, تم.

Four rules and why they work

Two — the last digit is even. Tens, hundreds and thousands are always divisible by two (10 = 2×5). What is left decides the remainder — the last digit. 734 ends with a four → divisible by two. Even numbers = divisible by two, it is the same thing.

Five — the last digit is 0 or 5. The same logic: tens can always be divided by five, the end decides. 120 ✓, 85 ✓, 82 ✗.

Ten — the last digit is 0. The strictest of the three: 120 ✓, 125 ✗. Notice: what is divisible by ten is automatically divisible by two and by five (10 = 2 × 5).

Three — the digit sum is divisible by three. Here the end of the number is not enough — three does not sit smoothly in the tens system (10 : 3 leaves remainder 1, each ten “smuggles” a one of remainder). So you add all the digits: 738 → 7+3+8 = 18 → 18 : 3 = 6 cleanly → 738 is divisible by three.

A detective’s method: look at the last digit (2, 5, 10), then add the digits (3). In a few seconds you know more about a number than a minute of dividing would tell you. And one number can happily pass more tests at once: 90 is divided by 2, 3, 5 and 10.

Check the number 234

Two: last digit 4 — even ✓.

Five: ends with 4, not 0 or 5 ✗.

Ten: does not end with zero ✗.

Three: 2 + 3 + 4 = 9. Nine can be divided by three ✓.

Verdict: 234 is divisible by 2 and 3 (and so by six too — who is divisible by two and three at once is divisible by their product). Check: 234 : 2 = 117 ✓, 234 : 3 = 78 ✓.

Check the number 605

Two: ends with five — odd ✗.

Five: ends with 5 ✓.

Ten: does not end with zero ✗.

Three: 6 + 0 + 5 = 11. Eleven cannot be divided by three (11 : 3 = 3, remainder 2) ✗.

Verdict: 605 is divisible only by five (of our four). Check: 605 : 5 = 121 ✓. A zero in the middle of the number has no effect on the rules — you count the last digit and the sum of all of them.

Simplifying the fraction 45/60 with divisibility

I look for common factors of the numbers 45 and 60 — that is exactly what the rules are for.

Five: 45 ends with five ✓, 60 with zero ✓ → I divide both by five: 45/60 = 9/12.

Three: 9 → ✓, 12 → 1+2 = 3 ✓ → I divide both by three: 9/12 = 3/4.

It goes no further (3 and 4 have no common factor). The fraction 45/60 = 3/4 — and the divisibility rules told me what to simplify by, without trying anything blind.

⚠️

The digit sum only works for three (and nine), not for the others. You cannot tell divisibility by two from the digit sum — 13 has sum 4, and still cannot be divided by two. Each divisor has its own rule: the end of the number for 2, 5, 10; the sum of the digits for 3.

Practice: now you do it 🎲

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

Decide about divisibility with no counting — only by the rules. Answer yes or no.

Practise the questions

On paper

Detective work: no dividing in columns, only the rules.

  1. Check the numbers 96, 155, 240 and 413: by each one write which of 2, 3, 5, 10 it is divisible by, and by which rule.

  2. Find all the numbers between 50 and 70 that are divisible by three. (Digit sums!)

  3. Fill in a digit instead of the star so that the number 52* is divisible by two and three at once. Find all the options.

  4. Simplify the fractions 30/45 and 24/60 — look for factors with the divisibility rules and write which rule you used.

Now you 💪

  1. I can tell divisibility by two, five and ten from the last digit.
  2. I can work out the digit sum and decide about divisibility by three.
  3. I use the rules when simplifying a fraction instead of trying blind.
✅

Done when: For any number up to a thousand you decide without dividing whether it is divisible by 2, 3, 5 and 10 — and you can use that when simplifying fractions.

What to take from this lesson

© 2026 Ing. Martin Polak / AlgoRhino · شروط استخدام المحتوى