Consecutive Integer Word Problem

[JNOD]

Question: "If N is any positive integer, how many consecutive integers following N are needed to ensure that at least one of the integers is divisible by another positive integer m?

Answer Choices:

(A) m - 1
(B) m
(C) m + 1
(D) 2m
(E) [imath]m^2[/imath]

Answer Key: (A): "One out of every m consecutive integers is divisible by the integer m. The wording of this question is difficult, so you might have thought the question was asking you to count N itself to get the correct answer. But it was not, so m - 1 consecutive integers after N plus N itself are required to ensure that one of these integers, including N, is the multiple of m.

Comment: Hello, this was one of the questions from a test prep. booklet I am currently using. I apologize if this question is in the wrong category. So, after re-reading this question several times, I still found it quite difficult to visualize. If anyone can explain the question in a more simpler, intuitive way, I'd greatly appreciate it.

My Approach:
Positive Integers: 1, 2, 3, ...
Consecutive numbers: x, x+1, x+2, x+3, ...
Divisibility definition: "the capacity of being evenly divided, without remainder."

So question states "if N is any positive integer..."
ok, so like 1, 2, 3

"how many consecutive numbers following N are needed..."
ok, so we're looking at consecutive numbers "following N" so (N+1), (N+2), (N+3)
I believe this is what one would consider "3 consecutive integers" : (N+1), (N+2), (N+3)

"how many [...] are needed to ensure that at least one of the integers is divisible by another positive integer m"
so, m could either be an odd or an even positive integer
a consecutive integer of N could start as an even or an odd integer

For example if N = 4

(N+1) = 5 (odd)
(N+2) = 6 (even)

Having two consecutive integers following 4 would yield at least one even and one odd integer. And if N was an odd integer to begin with, then the following two consecutive integers would also include an even and an odd integer.

So, if m is any positive integer, odd or even, how would I even know that it would be divisible by any odd or even number?
Am I interpreting this question correctly?

Looking at the answer key, I still don't really understand why "m - 1" is the number of consecutive integers following N.

Any insight and help is greatly appreciated.

 

[Deleted member 4993]

Question: "If N is any positive integer, how many consecutive integers following N are needed to ensure that at least one of the integers is divisible by another positive integer m?

Answer Choices:

(A) m - 1
(B) m
(C) m + 1
(D) 2m
(E) [imath]m^2[/imath]

Answer Key: (A): "One out of every m consecutive integers is divisible by the integer m. The wording of this question is difficult, so you might have thought the question was asking you to count N itself to get the correct answer. But it was not, so m - 1 consecutive integers after N plus N itself are required to ensure that one of these integers, including N, is the multiple of m.

Comment: Hello, this was one of the questions from a test prep. booklet I am currently using. I apologize if this question is in the wrong category. So, after re-reading this question several times, I still found it quite difficult to visualize. If anyone can explain the question in a more simpler, intuitive way, I'd greatly appreciate it.

My Approach:
Positive Integers: 1, 2, 3, ...
Consecutive numbers: x, x+1, x+2, x+3, ...
Divisibility definition: "the capacity of being evenly divided, without remainder."

So question states "if N is any positive integer..."
ok, so like 1, 2, 3

"how many consecutive numbers following N are needed..."
ok, so we're looking at consecutive numbers "following N" so (N+1), (N+2), (N+3)
I believe this is what one would consider "3 consecutive integers" : (N+1), (N+2), (N+3)

"how many [...] are needed to ensure that at least one of the integers is divisible by another positive integer m"
so, m could either be an odd or an even positive integer
a consecutive integer of N could start as an even or an odd integer

For example if N = 4

(N+1) = 5 (odd)
(N+2) = 6 (even)

Having two consecutive integers following 4 would yield at least one even and one odd integer. And if N was an odd integer to begin with, then the following two consecutive integers would also include an even and an odd integer.

So, if m is any positive integer, odd or even, how would I even know that it would be divisible by any odd or even number?
Am I interpreting this question correctly?

Looking at the answer key, I still don't really understand why "m - 1" is the number of consecutive integers following N.

Any insight and help is greatly appreciated.

Excellent thinking - but you got slightly disoriented. One of the key -words in the problem -statement is "at least" .
If you list a string of consecutive integers, every 2nd number will be divisible by 2
every 3rd number will be divisible by 3
every 4th number will be divisible by 4
and so on....
Now think a bit more.....

 

[Dr.Peterson]

Question: "If N is any positive integer, how many consecutive integers following N are needed to ensure that at least one of the integers is divisible by another positive integer m?

Answer Choices:

(A) m - 1
(B) m
(C) m + 1
(D) 2m
(E) [imath]m^2[/imath]

Answer Key: (A): "One out of every m consecutive integers is divisible by the integer m. The wording of this question is difficult, so you might have thought the question was asking you to count N itself to get the correct answer. But it was not, so m - 1 consecutive integers after N plus N itself are required to ensure that one of these integers, including N, is the multiple of m.

Comment: Hello, this was one of the questions from a test prep. booklet I am currently using. I apologize if this question is in the wrong category. So, after re-reading this question several times, I still found it quite difficult to visualize. If anyone can explain the question in a more simpler, intuitive way, I'd greatly appreciate it.

My Approach:
Positive Integers: 1, 2, 3, ...
Consecutive numbers: x, x+1, x+2, x+3, ...
Divisibility definition: "the capacity of being evenly divided, without remainder."

So question states "if N is any positive integer..."
ok, so like 1, 2, 3

"how many consecutive numbers following N are needed..."
ok, so we're looking at consecutive numbers "following N" so (N+1), (N+2), (N+3)
I believe this is what one would consider "3 consecutive integers" : (N+1), (N+2), (N+3)

"how many [...] are needed to ensure that at least one of the integers is divisible by another positive integer m"
so, m could either be an odd or an even positive integer
a consecutive integer of N could start as an even or an odd integer

For example if N = 4

(N+1) = 5 (odd)
(N+2) = 6 (even)

Having two consecutive integers following 4 would yield at least one even and one odd integer. And if N was an odd integer to begin with, then the following two consecutive integers would also include an even and an odd integer.

So, if m is any positive integer, odd or even, how would I even know that it would be divisible by any odd or even number?
Am I interpreting this question correctly?

Looking at the answer key, I still don't really understand why "m - 1" is the number of consecutive integers following N.

Any insight and help is greatly appreciated.

I think you are misunderstanding the problem. Let's take a specific example to see what it means.

Suppose m = 7. Then, putting that number into the problem,

If N is any positive integer, how many consecutive integers following N are needed to ensure that at least one of the integers is divisible by another positive integer, m = 7?​

The main idea here is that m is a given number, and N could be anything. (If we didn't know m, we could do nothing! And the answer choices wouldn't be expressions involving m!) On the other hand, if N were given, we could calculate an exact answer.

Furthermore, odd or even is irrelevant.

Suppose now that we pick N = 24. How many numbers following N do we need in order to include at least one number that is divisible by 7 (that is, is a multiple of 7)? Start counting: 25, 26, 27, 28, and we have a multiple of 7. So in this case, we had to take 4 consecutive numbers.

But if we don't know N ahead of time (imagine you're set down on the number line blindfolded!), what is the worst that can happen? How many steps do we have to take to be sure we have stepped on a multiple of 7?

For me, this sort of physical/visual modeling is the best way to see what to do.

But I think the problem is poorly worded, not just "difficult". I would have interpreted it as saying that we want one of the numbers following N to be a multiple of m, but their answer shows that they are including N among the numbers considered, since they say "m - 1 consecutive integers after N plus N itself are required". The problem failed to specify what set of numbers has to include a multiple of m.

 

[Steven G]

If you divide a number by say 7, the remainder can be 0, 1, 2, 3, 4, 5 or 6. If you have 7 consecutive numbers then one of them must be divisible by 7.

For example if the 1st of seven consecutive, m, has a remainder of 2 when divided by 7, then m+1 has a remainder of 3, m+2 has a remainder of 4, m+3 has a remainder of 5, m+4 has a remainder of 6,n m+5 has a remainder of 70, and m+6 has a remainder of 1.

 

[JNOD]

Thank you so much for your help, Subhotosh Khan, Dr.Peterson, and Steven G!

I was honestly stumped for a while, but your hints and insights have given me a fresh lens through which to look at this problem.

Initially, what I didn't understand was the connection between consecutive numbers and multiples, and thanks to Subhotosh Khan's and Steven G's hints, I gradually came to that realization, especially by employing Dr.Peterson's advice of physical/visual modeling.

After drawing up a random consecutive string of numbers, it all just suddenly clicked:

m-number of random consecutive numbers / m; m = 4

100/4 = 25
101/4 = 25 1/4
102/4 = 25 2/4
103/4 = 25 3/4

next m-number of consecutive numbers / m; m = 4

104/4 = 26
105/4 = 26 1/4
106/4 = 26 2/4
107/4 = 26 3/4

From this model, I see that there will always be at least 1 out of m consecutive integers that is divisible by m.

But, since the question is asking for numbers "following" N, the total number of m consecutive integers has to exclude 1, (m-1), so as to take the initial N into account.

 

[JNOD]

If you divide a number by say 7, the remainder can be 0, 1, 2, 3, 4, 5 or 6. If you have 7 consecutive numbers then one of them must be divisible by 7.

For example if the 1st of seven consecutive, m, has a remainder of 2 when divided by 7, then m+1 has a remainder of 3, m+2 has a remainder of 4, m+3 has a remainder of 5, m+4 has a remainder of 6,n m+5 has a remainder of 7, and m+6 has a remainder of 1.

Thank you, Steven G!

 

[Steven G]

Of course I meant to say that m+5 has a remainder of 0

 

[JNOD]

I think you are misunderstanding the problem. Let's take a specific example to see what it means.

Suppose m = 7. Then, putting that number into the problem,

If N is any positive integer, how many consecutive integers following N are needed to ensure that at least one of the integers is divisible by another positive integer, m = 7?​

The main idea here is that m is a given number, and N could be anything. (If we didn't know m, we could do nothing! And the answer choices wouldn't be expressions involving m!) On the other hand, if N were given, we could calculate an exact answer.

Furthermore, odd or even is irrelevant.

Suppose now that we pick N = 24. How many numbers following N do we need in order to include at least one number that is divisible by 7 (that is, is a multiple of 7)? Start counting: 25, 26, 27, 28, and we have a multiple of 7. So in this case, we had to take 4 consecutive numbers.

But if we don't know N ahead of time (imagine you're set down on the number line blindfolded!), what is the worst that can happen? How many steps do we have to take to be sure we have stepped on a multiple of 7?

For me, this sort of physical/visual modeling is the best way to see what to do.

But I think the problem is poorly worded, not just "difficult". I would have interpreted it as saying that we want one of the numbers following N to be a multiple of m, but their answer shows that they are including N among the numbers considered, since they say "m - 1 consecutive integers after N plus N itself are required". The problem failed to specify what set of numbers has to include a multiple of m.

Thank you, Dr.Peterson!

 

[JNOD]

Excellent thinking - but you got slightly disoriented. One of the key -words in the problem -statement is "at least" .
If you list a string of consecutive integers, every 2nd number will be divisible by 2
every 3rd number will be divisible by 3
every 4th number will be divisible by 4
and so on....
Now think a bit more.....

Thank you, Subhotosh Khan!

 

[JeffM]

I suggest you look back at the original problem. Your instinct that the answer key's answer of m - 1 is incorrect was right on the money.

As you have discovered, m divides evenly into exactly one of any consecutive sequence of m integers.

The question asks "how many consecutive integers following N are needed."

If N is divisible by m, then the next integer that follows N and is divisible by m is N + m. Thus, the sequence must start with N + 1 and run through N + m. That is m integers. The answer key tries to answer a different question. namely

Considering a sequence of consecutive integers starting with N, how many integers greater than N are required to ensure that exactly one integer in the sequence will be evenly divisible by positive integer m.

The question to that question is indeed m - 1 as the answer key explains. Unfortunately, that was not the question asked. Moreover, the use of the phrase "at least" is unfortunate. That allows for multiple instances of divisibility. Terrible problem.