What is the largest multiple of 12 that can be written using each digit 0, 1, 2, 3,........,9 exactly once?
SOURCE: Math Counts
Solution:
The problem asks for the largest number using the digits, and it also has to be divisible by 12.
So let's look at what is the largest number possible using the digits.
Given the place values the highest number has to come first. And second highest next and so on. So the highest number using the digits (without any conditions) will be 9876543210.
Next condition is that it has to be divisible by 12. We know that for a number to be divisible by 12, it has to be divisible by 3 and by 4.
Let's first check for divisibility by 3.
Divisibility Rule for 3: A number is divisible by 3 if the sum of the digits of that number is divisible by 3.
So we add up all the digits 9+8+7+6+5+4+3+2+1+0=45
45 is divisible by 3, so 9876543210 or any combination of that number* will also be divisible by 3*.
Next we need to see if that number is a multiple of 4.
Divisibility Rule for 4: For a number to be divisible by 4, the last 2 digits should be divisible by 4. So if we take 9876543210, the last 2 digits are 1 and 0 i.e. 10. 10 is not divisible by 4.
So we need to rearrange the numbers so that it becomes divisible by 4, but we still want the largest number possible.
So if we rearrange just the last 2 digits we get 9876543201 which is also not divisible by 4.
So let's rearrange the last 3 digits. The various combinations of the last 3 digits that are also divisible by 4 are 120 and 012 (the other combinations 021, 102, 210 and 201 are not divisible by 4). Out of those two number 120 is higher.
So the largest number that is divisible by 12 using all the digits exactly once is 9876543120.
Now try a problem on your own
What is the greatest 6 digit number that is a multiple of 12?
Post your answer in the comments below. The first person to give a correct answer to 5 questions will receive a free 30 min problem-solving session with me!
*If you have doubts on divisibility rule for 3 or need more examples and explanations, see the video below. Video for divisibility rule for 4 will be coming soon
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