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Let's look at a simple example first such as 3 consecutive integers. We want to show that it is divisible by 3.

Take 4,5, and 6 and of course 6 is divisible by 3.

The reason for this is can be seen using the pigeonhole principle.

When an integer is divided by 3, possible remainders are 0, 1, and 2. It follows that every

integer can be expressed in one of the forms 3k, 3k + 1, and 3k + 2 where k is an integer.

So if we have any three consecutive integers, one of them must be divisible by 3.

Let's look at how the pigeonhole principle applies. Suppose we have 3 consecutive integers are non are divisible by 3. Think of the pigeon holes as 3k, 3k + 1, and 3k + 2, now this means no numbers are in the 3k hole and two of them must be in either the 3k+1 hole or the 3k+2 hole. But this contradicts that they are consecutive integers.

So for any n, let the pigeon holes be nk, nk+1,... nk+(n-1) and these exhaust the multiples of n. Now if you take n consecutive numbers, you must have a least 1 number in the nk pigeon hole or else they are not consecutive.

Q: Use pigeonhole principle to show that one of any n consecutive integers divisible by n?

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6,12,18.

8

There aren't any. 340 is divisible by 4 while the sum of any two consecutive even numbers cannot be divisible by 4.There aren't any. 340 is divisible by 4 while the sum of any two consecutive even numbers cannot be divisible by 4.There aren't any. 340 is divisible by 4 while the sum of any two consecutive even numbers cannot be divisible by 4.There aren't any. 340 is divisible by 4 while the sum of any two consecutive even numbers cannot be divisible by 4.

consecutive integers

There is no set of six consecutive integers for -4.

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There must be three consecutive integers to guarantee that the product will be divisible by 6. For the "Product of three consecutive integers..." see the Related Question below.

6,12,18.

8

Not sure what thress is. If three, then there is no answer since the sum (or product) of any three consecutive integers must be divisible by 3.

yes always this is true' example 1,2,3 sum is 6 and is divisible by 3

Here are some consecutive odd integers. All must follow this form: x+x+2+x+2=3x + 6 6 is divisible by 3. So is 3x. If you add two integers that are divisible by 3, it is still divisible by three. The question is not about odd integers, but consecutive integers. This should be x + x+1 + x+2 =3x+3 A bit simpler would be the three in a row : x-1, x, x+1 which add up to 3x which can be divided by 3.

Any three consecutive integers are divisible by three because it can be shown that the sum divided by three is the middle number.

"Consecutive" integers are integers that have no other integer between them.

There aren't any. 340 is divisible by 4 while the sum of any two consecutive even numbers cannot be divisible by 4.There aren't any. 340 is divisible by 4 while the sum of any two consecutive even numbers cannot be divisible by 4.There aren't any. 340 is divisible by 4 while the sum of any two consecutive even numbers cannot be divisible by 4.There aren't any. 340 is divisible by 4 while the sum of any two consecutive even numbers cannot be divisible by 4.

Because 6*8 = 48 and 48/8 = 6

There are no "two consecutive integers" that can do that.But there are two consecutive even integers that can: 8 and 10 .

Three consecutive integers have a sum of 12. What is the greatest of these integers?