Most people feel that the topic of number properties is hard or at least a little tricky. The reason is that no matter how much effort you put into it, you will still come across new concepts every time you sit with some 700+ level problems of this topic. There will be some concepts you don’t know and will need to “figure out” during the actual test. I came across one such question the other day. It brought forth a concept I hadn’t thought about before so I decided to share it today:
Say you have N consecutive integers (starting from any integer). What can you say about their sum? What can you say about their product?
Say N = 3
The numbers are 5, 6, 7 (any three consecutive numbers)
Their sum is 5 + 6 + 7 = 18
Their product is 5*6*7 = 210
Note that both the sum and the product are divisible by 3 (i.e. N).
Say N = 5
The numbers are 2, 3, 4, 5, 6 (any five consecutive numbers)
Their sum is 2 + 3 + 4 + 5 + 6 = 20
Their product is 2*3*4*5*6 = 720
Again, note that both the sum and the product are divisible by 5 (i.e. N)
Say N = 4
The numbers are 3, 4, 5, 6 (any five consecutive numbers)
Their sum is 3 + 4 + 5 + 6 = 18
Their product is 3*4*5*6 = 360
Now note that the sum is not divisible by 4, but the product is divisible by 4.
If N is odd then the sum of N consecutive integers is divisible by N, but this is not so if N is even.
Why is this so? Let’s try to generalize – if we have N consecutive numbers, they will be written in the form:
(Multiple of N),
(Multiple of N) +1,
(Multiple of N) + 2,
… ,
(Multiple of N) + (N-2),
(Multiple of N) + (N-1)
In our examples above, when N = 3, the numbers we picked were 5, 6, 7. They would be written in the form:
(Multiple of 3) + 2 = 5
(Multiple of 3) = 6
(Multiple of 3) + 1 = 7
In our examples above, when N = 4, the numbers we picked were 3, 4, 5, 6. They would be written in the form:
(Multiple of 4) + 3 = 3
(Multiple of 4) = 4
(Multiple of 4) + 1 = 5
(Multiple of 4) + 2 = 6
etc.
What happens in case of odd integers? We have a multiple of N and an even number of other integers. The other integers are 1, 2, 3, … (N-2) and (N-1) more than a multiple of N.
Note that these extras will always add up in pairs to give the sum of N:
1 + (N – 1) = N
2 + (N – 2) = N
3 + (N – 3) = N
…
So when you add up all the integers, you will get a multiple of N.
What happens in case of even integers? You have a multiple of N and an odd number of other integers. The other integers are 1, 2, 3, … (N-2) and (N-1) more than a multiple of N.
Note that these extras will add up to give integers of N but one will be leftover:
1 + (N – 1) = N
2 + (N – 2) = N
3 + (N – 3) = N
…
The middle number will not have a pair to add up with to give N. So when you add up all the integers, the sum will not be a multiple of N.
For example, let’s reconsider the previous example in which we had four consecutive integers:
(Multiple of 4) = 4
(Multiple of 4) + 1 = 5
(Multiple of 4) + 2 = 6
(Multiple of 4) + 3 = 3
1 and 3 add up to give 4 but we still have a 2 extra. So the sum of four consecutive integers will not be a multiple of 4.
Let’s now consider the product of N consecutive integers.
In any N consecutive integers, there will be a multiple of N. Hence, the product will always be a multiple of N.
Now take a quick look at the GMAT question that brought this concept into focus:
Which of the following must be true?
1) The sum of N consecutive integers is always divisible by N.
2) If N is even then the sum of N consecutive integers is divisible by N.
3) If N is odd then the sum of N consecutive integers is divisible by N.
4) The Product of K consecutive integers is divisible by K.
5) The product of K consecutive integers is divisible by K!
(A) 1, 4, 5
(B) 3, 4, 5
(C) 4 and 5
(D) 1, 2, 3, 4
(E) only 4
Let’s start with the first three statements this question gives us. We can see that out of Statements 1, 2 and 3, only Statement 3 will be true for all acceptable values of N. Therefore, all the answer choices that include Statements 1 and 2 are out, i.e. options A and D are out. The answer choices that don’t have Statement 3 are also out, i.e. options C and E are out. This leaves us with only answer choice B, and therefore, B is our answer.
This question is a direct application of what we learned above so it doesn’t add much value to our learning as such, but it does have an interesting point. By establishing that B is the answer, we are saying that Statement 5 must be true.
5) The product of K consecutive integers is divisible by K!
We will leave it to you to try to prove this!
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