The Genius Bank Robber

Question

StackVille Bank & Trust - Robbed!

The Robbery

The bank is being robbed. There are millions in cash in the vault. The lone robber takes the manager to the safe and demands he opens it.

The manager was scared, as he did not know the three numbers to the vault. He did know, however, where the bank president kept clues to the combination. He opened a drawer, pulled out a piece of paper, and read the clues:

The Combination

1. If the second number was not a multiple of 8, then it was a number from 80 through 89.

2. If 1st number was not a multiple of 3, then it was a number from 60 through 69.

3. If the third number was a multiple of 3, then it was a number from 50 through 59.

4. If the first number was a multiple of 2, then it was a number from 50 through 59

5. If the second number was a multiple of 6, then it was a number from 40 through 49.

6. If 2nd number was not a multiple of 7, then it was a number from 60 through 69.

7. If the first number was not a multiple of 4, then it was a number from 70 through 79.

8. If 3rd number was not a multiple of 4, then it was a number from 60 through 69.

9.If the third number was not a multiple of 6, then it was a number from 70 through 79.

The vault safe numbers range from 50-79.

The Get-A-Way

The robber looked at the manager, smiled and said, "Thanks!"

He opened the vault, grabbed the cash, and made a smooth escape. The robber figured out the vault's combo - can you?

What is the vault combination?

Answer 1

Rearrange:

• 2. If 1st number was not a multiple of 3, then it was a number from 60 through 69.
• 4. If the first number was a multiple of 2, then it was a number from 50 through 59
• 7. If the first number was not a multiple of 4, then it was a number from 70 through 79.
• 1. If the second number was not a multiple of 8, then it was a number from 80 through 89.
• 5. If the second number was a multiple of 6, then it was a number from 40 through 49.
• 6. If 2nd number was not a multiple of 7, then it was a number from 60 through 69.
• 3. If the third number was a multiple of 3, then it was a number from 50 through 59.
• 8. If 3rd number was not a multiple of 4, then it was a number from 60 through 69.
• 9. If the third number was not a multiple of 6, then it was a number from 70 through 79.

The answer is:

75 - (56 or 64) - 76

As:

If first number is a multiple of 2 but not a multiple of 4 (e.g. 2, 6, 10, ...), then it is both between 50-59 and between 70-79. This cannot be true, so one of the following is true

* First number is multiple of 4 and is between 50-59: 52, 56
* First number is not a multiple of 2 and is between 70-79: 71, 73, 75, 77, 79

If first number is not a multiple of 3, then it is between 60-69, which is not true, so first number is a multiple of 3.

=> First number is 75

If second number is a multiple of 6, then it is between 40-49: 48
Then it is not a multiple of 7, so it must be between 60-69, which cannot be true, so second number is not a multiple of 6.

Second number cannot be simultaneously between 80-89 and between 60-69, so it cannot be simultaneously not a multiple of 7 and not a multiple of 8.
Option 1: Not a multiple of 7, but a multiple of 8: then it is between 60-69, so 64 (which is not a multiple of 6)
Option 2: A multiple of 7, but not a multiple of 8: then it is between 80-89, so 84, but this is a multiple of 6, so this option is false.
Option 3: A multiple of 7 and a multiple of 8: 56, which is not a multiple of 6

=> Second number is 56 or 64

If third number is a multiple of 3 but not a multiple of 6 (3, 9, 15, ...), then it is both between 50-59 and between 70-79. This cannot be true so one of the following is true:
* Third number is multiple of 6 and is between 50-59: 54
* Third number is not a multiple of 3 and is between 70-79: 70, 71, 73, 74, 76, 77, 79

If third number is not a multiple of 4, then is between 60-69, which is not true, so third number is a multiple of 4.

=> Third number is 76

Answer 2

The code is

75-64-76

---

Looking at the individual numbers:

For the first number:

If it's not a multiple of 3, then its 60-69
If it's a multiple of 2, then it's 50-59
If it's not a multiple of 4, then it's 70-79

So:

If it's 50-59, it's a multiple of 2, 3 and 4; the number must be a multiple of 12. No multiples of 12 fall in the range 50-59.

If it's 60-69, it's not a multiple of 3, not a multiple of 2 and a multiple of 4. As all multiples of 4 are also multiples of 2, this cannot be.

If it's 70-79, then it's not a multiple of 4, not a multiple of 2 and a multiple of 3. Multiples of 3 in this range are 72, 75 and 78. 75 fits, so the first number is 75.

For the second number:

If it's not a multiple of 8, then it's 80-89
If it's a multiple of 6, then it's 40-49
If it's not a multiple of 7, then it's 60-69

So:

If it's 40-49, it's a multiple of 8, a multiple of 7 and a multiple of 6. No such number exists.

If it's 60-69, it's not a multiple of 7, not a multiple of 6 and a multiple of 8. Multiples of 8 in this range are 64, which fits. So the second number is 64.

However

As @Shoover and @Magma points out, 56 also works, as do an infinite amount of numbers that are both a multiple of both 7 and 8, and not a multiple of 6. The OP has clarified in the comments under this answer that the number must be in the ranges given, meaning it is 64.

Finally, for the third number (initially typed a statement wrong):

If it's a multiple of 3, then it's 50-59
If it's not multiple of 4, then it's 60-69
If it's not a multiple of 6, then it's 70-79

So:

If it’s 50-59, then it’s a multiple of 3, 4 and 6. Multiples of 6 in this range are 54, which is not divisible by 4, so it’s not that.

If it’s 60-69, then it’s not a multiple of 3 or 4, and a multiple of 6. All multiples of 6 are also multiples of 3 so this cannot be.

If it's 70-79, then it's a not a multiple of 6 that's a multiple of 4 and not a multiple of 3. Multiples of 4 in this range are 72, 76 - only 76 fits here, so it’s 76.

Which gives us

75-64 - 76