Lesson: Consecutive Integers

Be careful. abc does not represent the product of three numbers.

Each variable (a, b, and c) represents a DIGIT in the 3-digit number abc. As such the rules related to the product (n-1)n(n+1) do not hold.

Yes, on test day, the question will explicitly state the kinds of consecutive numbers.
Consecutive integers ...-3, -2, -1, 0, 1, 2, 3, ...
Consecutive EVEN integers: ...-6, -4, -2, 0, 2, 4, 6, ...
Consecutive ODD integers: ...-5, -3, -1, 1, 3, 5,...

Question Link: https://gmatclub.com/forum/if-n-is-a-positive-integer-is-n-3-n-divisible...

You are correct; 0 IS divisible by 4 (in fact, 0 divisible by all integers)

If x and y are integers, we can say that x is divisible by y, if we can write x = yk, where k is some integer.

For example, 12 is divisible by 3, because we can write 12 = (3)(4), and 4 is an integer.
And 70 is divisible by 5, because we can write 70 = (5)(14), and 14 is an integer.
Likewise, 0 is divisible by 0, because we can write 0 = (4)(0), and 0 is an integer.

Does that help?

Cheers,
Brent

Yes, that's correct. However, I've never seen an official GMAT question that requires this property. In fact, the GMAT test-makers typically restrict the values in Integer Properties question to POSITIVE integers only.

Question link: https://gmatclub.com/forum/the-sum-of-4-different-odd-integers-is-64-wha...

Statement 1: The integers are consecutive odd numbers
Let's examine some consecutive odd numbers: 1, 3, 5, 7, 9, 11, 13....

Notice that each integer is 2 greater than the number before it.

So, if we let x = the first number...
then x + 2 must be the next integer
and x + 2 + 2 must be the integer after that.
And so on.

Does that help?

Cheers,
Brent

Here's my full solution: https://gmatclub.com/forum/the-sum-of-4-different-odd-integers-is-64-wha...

Bunuel's rule doesn't contradict the rule in the video lesson. In fact, his law is very similar to mine.

Keep in mind that, within the product of n consecutive integers, we also have the product of n-1 consecutive integers.

For example, consider the product of 5 consecutive integers: (3)(4)(5)(6)(7)
This product must be divisible by 5

Also, within these 5 consecutive integers, we have the product of 4 consecutive integers, (3)(4)(5)(6), which means (3)(4)(5)(6) must be divisible by 4, which means (3)(4)(5)(6)(7) is divisible by 4 as well.

Likewise, within these 5 consecutive integers, we have the product of 3 consecutive integers, (3)(4)(5), which means (3)(4)(5) must be divisible by 3, which means (3)(4)(5)(6)(7) is divisible by 3 as well.

And so on.

So.....

Let's say K = the product of n consecutive integers.
As such, K must be divisible by n

Also, within the n consecutive integers, there are n-1 consecutive integers.
As such, K must be divisible by n-1

Also, within the n consecutive integers, there are n-2 consecutive integers.
As such, K must be divisible by n-2

etc

So, K must be divisible by n, n-1, n-2, n-3, . . . . 2, 1
In other words, K must be divisible by n!

Cheers,
Brent

0 is a multiple of all integers.
That said, pretty much all official GMAT questions involving integer properties will limit the values to positive integers.

Here's one approach: https://gmatclub.com/forum/what-is-the-sum-of-the-integers-from-100-to-2...

Cheers,
Brent

Link: https://gmatclub.com/forum/if-n-is-a-positive-integer-is-n-3-n-divisible...

There's no rule that says (k+1)(k)(2k+1) is DIVISIBLE by 4
Notice that, if k = 2, then (k+1)(k)(2k+1) = (3)(2)(5) = 30, and 30 isn't divisible by 4.

There is a rule that says the product of n consecutive integers MUST be divisible by n, n-1, n-2,..., 2 and 1
For example, the product of 6 consecutive integers MUST be divisible by 6, 5, 4, 3, 2 and 1

Likewise, the product of 3 consecutive integers MUST be divisible by 3, 2 and 1
If the product is divisible by 3 and 2, the product is also divisible by 6.

Cheers,
Brent

Yes, if there are two even numbers as part of a product, then that product must be divisible by 4.

If we CAN take a set of integers and arrange those integers in ascending order so that each integer is one greater than the integer before it, then we have a set of consecutive integers.

So, for example {4, 7, 6, 5} is a set of consecutive integers.

I mention this because I've seen questions that state x, y and z are consecutive integers. In these cases, we can't immediately assume that x < y < z.
It COULD be the case that x = 4, y = 2 and z = 3.

Cheers,
Brebt