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E
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
45%
(medium)
Question Stats:
65% (01:35) correct
35%
(01:16)
wrong
based on 629
sessions
History
Date
Time
Result
Not Attempted Yet
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
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).
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).
11 Jun 2012, 20:27
Kudos
Bookmarks
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)
Regards,
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)
Regards,
General Discussion
11 Jun 2012, 20:13
Kudos
Bookmarks
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.
