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It is known that no more than 7 children will be attending a party. Wh

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It is known that no more than 7 children will be attending a party. Wh [#permalink] New post   Updated on: 05 Feb 2019, 07:31
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It is known that no more than 7 children will be attending a party. What is the smallest number of cookies that must be brought to the party so that each child receives the same number of cookies?

A. 35
B. 105
C. 180
D. 210
E. 420

Show SpoilerMY DOUBT
** Can someone please explain to me the intuition behind this answer? I figure it's a very straightforward answer, but what I don't understand is assuming the maximum number of children attend the party which is 7 and among the answers 35 looks to suffice as each child will receive 5 cookies, which makes it the smallest number among the answers. I'm a bit confused can someone point me to the right direction, Thanks. **


Show SpoilerSOLUTION

Since the number of cookies must be divisible by 1, 2, 3, 4, 5, 6, and 7, let's find the least common multiple of the integers 1, 2, 3, 4, 5, 6, and 7. Every integer is divisible by 1. Let's find the prime factorizations of 2, 3, 4, 5, 6, and 7.

So we want to find the least common multiple of these integers, which we will write in a column.

The largest number of times the prime factor 2 appears in any of these integers is 2, in . The largest number of times the prime factor 3 appears in any of these integers is 1. Similarly, the largest number of times the prime factors 5 appears in any of these integers is 1, and the largest number of times the prime factors 7 appears in any of these integers is 1.

So the least common multiple of the first 7 positive integers must contain 2 prime factors of 2, 1 prime factor of 3, 1 prime factor of 5, and 1 prime factor of 7.

The least common multiple of the first 7 positive integers is 2 × 2 × 3 × 5 × 7 = 420.

The smallest number of cookies that must be brought to the party is 420, choice (E).
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E

Originally posted by iNumbv on 11 Jun 2012, 13:53.
Last edited by Bunuel on 05 Feb 2019, 07:31, edited 3 times in total.
Edited the question and moved to PS forum.
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Re: It is known that no more than 7 children will be attending a party. Wh [#permalink] New post   11 Jun 2012, 20:27
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Hi Unan,

The question states that there are no more than 7 children in the party. Thus, the possibility is that there could be 1, 2, 3, 4, 5, 6 or 7 children.

If you assume answer as 35 and there are 3 children, you may not distribute be able to distribute cookies equally.
similarly if there were 105 cookies, and 2 children, cookies cannot be distributed equally.
or if there were 210 cookies, and 4 children, cookies cannot be distributed equally.

Thus, the question asks for a number of cookies which can be distributed to any number of children (from 1 to 7).

And therefore the smallest number of cookies would be lcm of (1, 2, 3, 4, 5, 6, 7), i.e., 420.

Answer (E)

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Re: It is known that no more than 7 children will be attending a party. Wh [#permalink] New post   11 Jun 2012, 20:13
I'm reasonably certain that the problem here is that the question is just a little poorly-worded. My assumption is that they want you to give out all of the cookies, in which case the answer is as you posted. If it were not necessary to use all of the cookies, then the answer would actually just be 7, assuming that each child has to get at least 1 whole cookie.
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Re: It is known that no more than 7 children will be attending a party. Wh [#permalink] New post   12 Jun 2012, 09:38
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I just took a simpler route.

Because there are no more than 7 children # of cookies has to be divisible by 7.

Question does not tell us how many children are going to show up it has to be divisible by 6 as well. And 5....1.
So what I did was take 6*7=42 and the answer has to be divisible by 5, 4, 3 ,2 1 there is only one answer choice that satisfies these constraints which is 420



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Re: It is known that no more than 7 children will be attending a party. Wh [#permalink] New post   17 Jun 2012, 12:18
I also took the simpler route. Found the answer that was divisible by 1 thru 7.

Question: Why did you multiply 6*7? Did you look for an answer that was also divisible by 42?
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Re: It is known that no more than 7 children will be attending a party. Wh [#permalink] New post   17 Jun 2012, 20:13
I agree with the last 2 responses in terms of how to attack the problem. Do not get caught up with the story of the problem. This type of problem is shown in the OG problem solving book with several variations. It all comes down to answering what number is divisible by the constraint (in this case the numbers 1 through 7).
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It is known that no more than 7 children will be attending a party. Wh [#permalink] New post   24 Jul 2016, 10:58
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Bunuel wrote:
It is known that no more than 7 children will be attending a party. What is the smallest number of cookies that must be brought to the party so that each child receives the same number of cookies?

A. 35
B. 105
C. 180
D. 210
E. 420


No more than 7 children attending the party means the number of children could be 1,2,3,4,5,6,7.

We need to find a number that is divisible by each of these. Out of the choices given only 420 is divisible by each of these numbers.

Hence , answer is E.
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Re: It is known that no more than 7 children will be attending a party. Wh [#permalink] New post   22 Sep 2017, 20:06
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The question states that there are no more than 7 children in the party. Thus, the possibility is that there could be 1, 2, 3, 4, 5, 6 or 7 children.
Cookies are to be equally divided among the children attending party.
Thus, the question asks for a number of cookies which can be distributed to any number of children (from 1 to 7).

And therefore the smallest number of cookies would be lcm of (1, 2, 3, 4, 5, 6, 7), i.e., 420.

Answer (E)
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Re: It is known that no more than 7 children will be attending a party. Wh [#permalink] New post   10 Jul 2020, 10:41
iNumbv wrote:
It is known that no more than 7 children will be attending a party. What is the smallest number of cookies that must be brought to the party so that each child receives the same number of cookies?

A. 35
B. 105
C. 180
D. 210
E. 420

Show SpoilerMY DOUBT
** Can someone please explain to me the intuition behind this answer? I figure it's a very straightforward answer, but what I don't understand is assuming the maximum number of children attend the party which is 7 and among the answers 35 looks to suffice as each child will receive 5 cookies, which makes it the smallest number among the answers. I'm a bit confused can someone point me to the right direction, Thanks. **


Show SpoilerSOLUTION

Since the number of cookies must be divisible by 1, 2, 3, 4, 5, 6, and 7, let's find the least common multiple of the integers 1, 2, 3, 4, 5, 6, and 7. Every integer is divisible by 1. Let's find the prime factorizations of 2, 3, 4, 5, 6, and 7.

So we want to find the least common multiple of these integers, which we will write in a column.

The largest number of times the prime factor 2 appears in any of these integers is 2, in . The largest number of times the prime factor 3 appears in any of these integers is 1. Similarly, the largest number of times the prime factors 5 appears in any of these integers is 1, and the largest number of times the prime factors 7 appears in any of these integers is 1.

So the least common multiple of the first 7 positive integers must contain 2 prime factors of 2, 1 prime factor of 3, 1 prime factor of 5, and 1 prime factor of 7.

The least common multiple of the first 7 positive integers is 2 × 2 × 3 × 5 × 7 = 420.

The smallest number of cookies that must be brought to the party is 420, choice (E).



Given: It is known that no more than 7 children will be attending a party.
Asked: What is the smallest number of cookies that must be brought to the party so that each child receives the same number of cookies?

Prime factorisation of 1,2,3,4,5,6,7
1 = 1
2 = 2
3 = 3
4= 2^2
5 = 5
6 = 2*3
7 = 7
Since we don't know the exact number of children attending the party, we have to bring number of cookies which will be equally divided irrespective of number of children attending the party. Number of children = {1,2,3,4,5,6,7}

2^2*3*5*7 = 420

Now the cookies can be divided equally among children irrespective of number of children attending the party.

IMO E
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Re: It is known that no more than 7 children will be attending a party. Wh [#permalink] New post   03 Dec 2021, 19:41
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Re: It is known that no more than 7 children will be attending a party. Wh [#permalink]
03 Dec 2021, 19:41
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