**Difficulty: Extreme**

A teacher wrote a whole number on the whiteboard and asked her students to say what numbers it could divide by.

The 1st student said, “the number is divisible by 1.”

The 2nd student said, “the number is divisible by 2.”

The 3rd student said, “the number is divisible by 3.”

And this continued until the 7th student said, “the number is divisible by 7.”

The teacher said only two students spoke incorrectly and these students spoke one after the other (consecutively).

What are the two incorrect numbers?

**Scroll down or click for a hint, or the answer!
**

## Brainteaser hint

The two students who spoke incorrectly were one after the other – so one of them said an odd number and the other said an even one.

For a bigger hint:

If a number is divisible by 2 and also by 3, then it will be divisible by 6 (because 2 x 3 = 6).

## Brainteaser answer

This brainteaser is really tricky, so it’s okay to get a bit stuck.

We know that the 2 incorrect numbers are consecutive. We can start by writing down all the pairs of consecutive numbers:

(1,2) (2,3) (3,4) (4,5) (5,6) (6,7)

And try to eliminate them.

Every whole number is divisible by 1, so 1 can’t be incorrect. We can eliminate (1,2).

If 2 is incorrect, the number on the board is odd. And that means 4 and 6 are also incorrect (odds aren’t divisible by even numbers). That’s 3 incorrect answers, which breaks the rules, because there are only supposed to be 2 incorrect answers. We can eliminate (2,3) since 2 must be correct.

If 3 is incorrect, then there’s no way the number can be divisible by 6. That breaks the rules, because the incorrect answers are supposed to be consecutive (3 and 6 are not consecutive). We can eliminate (3,4) since 3 must be correct.

Since both 2 and 3 are correct, the number on the board will also be divisible by 6 (because 2 x 3 = 6). This means 6 is also a correct answer we can rule out.

Now, we have ruled out 1, 2, 3 and 6. The only consecutive pair left is 4 and 5. So that’s our answer!

For a final check, let’s try and make up a number that could go on the board.

To start with, 6 is a natural choice. It is divisible by 1, 2, 3 and 6, and it’s not divisible by 4 or 5. Unfortunately, it’s also not divisible by 7.

What’s an easy number that’s divisible by 6 and 7? 6 x 7 = 42.

If we divide it out we get:

42 ÷ 1 = 42

42 ÷ 2 = 21

42 ÷ 3 = 14

**42 ÷ 4 = 10.5**

**42 ÷ 5 = 8.4**

42 ÷ 6 = 7

42 ÷ 7 = 6

It worked! 4 and 5 must be the two incorrect numbers.

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