The die is cast: The Wknd Puzzle by Dilip D’Souza

Use a die to generate different six-digit numbers. Can you figure out how many different six-digit numbers you could generate that are divisible by 6?

Your friend Abhilasha is something of a mathematics connoisseur. One day she runs into you and when you’ve both picked yourselves up, she hands you a die. You know, one of those small cubes with the numbers 1, 2, 3, 4, 5, 6 on the sides.

“Go home and do this,” she instructs you. “Toss the die and write down the number on top. Do this six times so you get a six-digit number - except, if you get a number you’ve already tossed, forget it and toss again. That is, I want you to generate a six-digit number whose digits are all different. For example, 415263. Or 361245.”

“OK,” you say. “Then what?”

“Keep doing this, generating six-digit numbers and writing them down.”

“OK,” you say. “So?”

“So here’s my question: how many different six-digit numbers will you generate that are divisible by 6?”

You go home. You’re grateful for the die, but you quickly realize you don’t really have to toss it hundreds and thousands of times. Can you answer Abhilasha’s question?

Hint: First, figure out how many different six-digit numbers, with no repetition of digits, you will generate with the die.

There are six possibilities for the first digit. For each of those, there are five possibilities for the second digit. For each pair of first-and-second digits, there are four possibilities for the third. And so on.

Thus there are 6 x 5 x 4 x 3 x 2 x 1 = 720 possible six-digit numbers with no digits repeated.

Does that help?

Scroll down for the answer.

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Answer:

Note that all of the 720 numbers are multiples of 3. How do we know this? Because in each case, the digits add up to 21 (6 + 5 + 4 + 3 + 2 + 1), which is a multiple of 3. And any number whose digits add to a multiple of 3 is itself a multiple of 3.

Now note that any multiple of 3 is also a multiple of 6 if it is even. In this case, your six-digit number will be even if it ends in 2, 4 or 6. Half of the six-digit numbers end like that - the others end in 1, 3, or 5.

That is, half of the generated numbers are even. There are 720 altogether. Thus the number that are multiples of 6 is 360.