The question is in the information below...

When the King of Vortzegia found that there was a petrol shortage, he introduced a petrol rationing system according to the 3-digit number on a car's number plate

People could only buy petrol on:

Monday: If the three digit number was odd

Tuesday: If the Digit sum of the number was greater than ten

Wednesday: If the number was a multiple of 3

Thursday: If the digit sum of the number was less than 15

Friday: if at least 2 of the 3 digits were the same

Saturday: if the three digit number was less than 500

Sunday: if each of the digits was less than six

Questions:

(A) On which days could a car with the number 726 be filled up with petrol?

(B) The king's limo had a number plate that could be filled up everyday. What was the number on it?

### 3 Answers | Add Yours

**A) The car number is 726**

It is even - **can't fill up on Monday**

The Digit sum = 7+2+6 = 15

It is greater than 10, - **can fill up on Tuesday**

726/3 = 242

It is a multiple of 3. - **can fill up on Wednesday**

Digit sum is not less than 15

**can't fill up on Thursday**

No two digits are same - so **can't fill up on Friday**

It is not less than 500 - So **can't fill up on Saturday**

Two digits are not less than 6- So **can't fill up on Saturday**

So the days the car can fill up on are **Tuesday and Wednesday only**.

**B) King's car. **

The king's number should satisfy all the requirments.

If the three digit number is less than 500, the maximum digit it can get for the hundreds place is 4. (either 4,3,2,1)

If each of the digit is less than 6, then tens place and ones place can have any of the following (5,4,3,2,1,0)

The number must be odd, therefore the ones place can only have 5 or 3 or 1.

Therefore all the combinations possible are,

4 5 5

3 4 3

2 3 1

1 2

1

0

Also it should be a multiple of 3 which means their sum of digits must be also divisible by 3. But we know their sum of digits mut be either 11,12,13 or 14. Therefore the sum of digits must be 12.

There is also another rule which says at least two digits must be equal. That means if two are equal the other must be an even number since the sum is 12, which is even

All the possible combinations for these requirements are,

That means the hundreds place and the tens place can't carry same number, which would result in an even number.

Therefore it should be either hundreds place and ones place or tens place or ones place

The possible numbers for ones place are 1, 3, 5

But the equal digits cannot be 1 or 3, since the sum should be 12. If the two digits is 3, the remaining digit would be 6. Therefore, the equal digit is 5.

**The number of the king's car is 255.**

i agree with the answer but im not sure if i can ask an extension question here but, here it is: the king wanted the queen to fill up her car on 6 days. What number should appear on the numberplate. Provide all possible answers.

This is a maths challenge problem in Victoria Australia.

The rules of the Challenge forbid online resources and all answers must be your own work.

Don't cheat!

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