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13 Apr 2020, 10:26
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A
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Difficulty:
75%
(hard)
Question Stats:
63% (02:57) correct
37%
(03:23)
wrong
based on 202
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In a certain school, a teachers wants to divide students in groups, so that each group has equal number of students. If the teacher divide the students in groups of 16 per group, then there will be 12 students left; if the teacher divide the students in groups of 24 per group, then there will be 20 students left; if the teacher divide the students in groups of 25 per group, then there will be 21 students left and if the teacher divide the students in groups of 30 per group, then there will be 26 students left. What is the minimum number of students that could be in the school?
A. 1196
B. 1200
C. 1216
D. 1784
E. 2396
Are You Up For the Challenge: 700 Level Questions
A. 1196
B. 1200
C. 1216
D. 1784
E. 2396
Are You Up For the Challenge: 700 Level Questions
14 Apr 2020, 02:00
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When you closely observe the various divisors and the respective remainders, you notice one important fact – the difference between the divisors and the remainders is constant in each case.
When you observe this in a question, note that the number (in this case, the number of students) can be expressed as follows:
Number, N = LCM (divisors)k – (common difference), where k is a positive integer.
In this question, the divisors are 16, 24, 25 and 30. The respective remainders are 12, 20, 21 and 26. The common difference between the divisors and the remainders is 4.
Therefore, number of students = LCM(16,24,25,30)k – 4.
LCM(16, 24, 25,30) = 1200.
Therefore, number of students = 1200k – 4.
Since we want the minimum number of students, we take the smallest value for k i.e. k =1. Minimum number of students = 1200 – 4 = 1196.
The correct answer option is A.
Hope that helps!
When you observe this in a question, note that the number (in this case, the number of students) can be expressed as follows:
Number, N = LCM (divisors)k – (common difference), where k is a positive integer.
In this question, the divisors are 16, 24, 25 and 30. The respective remainders are 12, 20, 21 and 26. The common difference between the divisors and the remainders is 4.
Therefore, number of students = LCM(16,24,25,30)k – 4.
LCM(16, 24, 25,30) = 1200.
Therefore, number of students = 1200k – 4.
Since we want the minimum number of students, we take the smallest value for k i.e. k =1. Minimum number of students = 1200 – 4 = 1196.
The correct answer option is A.
Hope that helps!
General Discussion
13 Apr 2020, 10:39
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Bunuel
Given:
1. In a certain school, a teachers wants to divide students in groups, so that each group has equal number of students.
2. If the teacher divide the students in groups of 16 per group, then there will be 12 students left;
3. if the teacher divide the students in groups of 24 per group, then there will be 20 students left;
4. if the teacher divide the students in groups of 25 per group, then there will be 21 students left and
5. if the teacher divide the students in groups of 30 per group, then there will be 26 students left.
Asked: What is the minimum number of students that could be in the school?
A. 1196 = 74*16+12 = 49*24+20 = 47*25+21 = 30*39+26
B. 1200 INCORRECT Since 1200mod30 <>26
C. 1216 INCORRECT Since 1216mod30 <>26
D. 1784 INCORRECT Since 1784mod30 <>26
E. 2396
IMO A
