# Is 3 divisible by 3

### Mysterious three

Tamme is in class. He looks at the clock and waits for it to ring. It is **12:45 p.m.**. Time does not pass. (Are you familiar with that? :) )

Tamme thinks about the clock: Funny - are all the numbers divisible by three on the clock?

3,6, 9 and 12 are divisible by 3.

Next: 15 and 30 are also divisible by 3. 45 too? It is more difficult. 45 is 30 plus 15. Then 45 is also divisible by 3. Can you find out more easily?

He thinks: Neither 4 nor 5 are divisible by 3. Suddenly he has an idea, he adds the digits: $$ 4 + 5 = 9 $$

This goes through 3. Wow! Does that mean when you add up the digits, do you see if a number is divisible by 3?

If you add up the digits of a number, that's it **Checksum** the number.

Example:

The checksum of 126 is 9 because $$ 1 + 2 + 6 = 9 $$.

### Tamme is in trouble

The teacher thinks Tamme is dreaming and calls: "Now it hits 13".

Tamme answers, completely absorbed in his numbers:

“$$ 13 cdot 3 = 39 $$. So 39 is divisible by 3. The cross sum of 39: $$ 3 + 9 = 12 $$. 12 is divisible by 3, and so is 39.

That's great. You only need to add the digits and you know immediately whether a number is divisible by 3 or not. "

A number is divisible by 3 if its sum is divisible by 3.

The teacher is thrilled that Tamme is thinking about numbers and math!

He asks Tamme: "Is 5931 divisible by 3?"

Tamme does the math: The checksum of 5931 is 18, because: $$ 5 + 9 + 3 + 1 = 18 $$. 18 is divisible by 3, so 5931 is also divisible by 3.

Tamme does the math in writing: 5931: 3 = 1977, without remainder.

### How about the 6 or 9?

In the afternoon, Tamme continues pondering: Does the rule also work with the 6 or 9?

Tamme collects in a table:

number | Cross sum | divisible by 6 | divisible by 9 |
---|---|---|---|

$$18$$ | $$1+8=9$$ | yes, $$ 3 cdot 6 = 18 $$ | yes, $$ 2 cdot 9 = 18 $$ |

$$21$$ | $$2+1=3$$ | No | No |

$$24$$ | $$2+4=6$$ | yes, $$ 4 cdot 6 = 24 $$ | No |

$$27$$ | $$2+7=9$$ | No | yes, $$ 3 cdot 9 = 27 $$ |

- A number is divisible by 6 if it is even and its cross sum is divisible by 3.

Example: 24 is divisible by 6, because 24 is even and the checksum is 6. 6 is divisible by 3. - A number is divisible by 9 if its sum is divisible by 9.

Example: 27 is divisible by 9 because the checksum of 27 is 9. 9 is divisible by 9.

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### Number puzzle

Tamme is pretty happy with what he found out. At the end he asks himself a riddle: "Can I change the number 49231 so that it is divisible by 3 and 6 and 9?"

Let's go:

“The number should be divisible by 6, so it has to be even and divisible by 3. If the number is divisible by 9, it is also divisible by 3.

That means: I need an even number whose sum is divisible by 9.

The cross sum of 49231 is 4 + 9 + 2 + 3 + 1 = 19. So I'm looking for a cross sum near 19 that is divisible by 9. That is 27. From 19 to 27 the difference is 8.

I have to change the digits so that the cross sum comes out to 27 and the last digit has to be even.

For example: 49248 or 79236. There are many options. "

Any number divisible by 6 or 9 is also divisible by 3.

### Divisibility rules at a glance

These are the divisibility rules for 3, 6 and 9.

Divisibility | rule | example |
---|---|---|

divisible by 3 | A number is divisible by 3 if its sum is divisible by 3. | The checksum of 39 is 12. The checksum of 12 is 3. So 39 is divisible by 3. |

divisible by 6 | A number is divisible by 6 if the number is even (divisible by 2) and its checksum is divisible by 3. | The checksum of 42 is 6. 6 is divisible by 3. The last digit of 42 is even. So 42 is divisible by 6. |

divisible by 9 | A number is divisible by 9 if its sum is divisible by 9. | The checksum of 108 is 9. 9 is divisible by 9. So 108 is even divisible by 9. |

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