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07 Dec 2020, 08:00
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C
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Difficulty:
25%
(medium)
Question Stats:
79% (01:49) correct
21%
(01:39)
wrong
based on 85
sessions
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A group of friends on a hoke split up into smaller groups. When they formed groups of three, one person was left out. When they formed groups of four, two people were left out. When they formed groups of six, four people were left out. If the group contains at least a dozen people, what is the smallest possible number of people in the group?
A. 10
B. 20
C. 22
D. 26
E. 34
A. 10
B. 20
C. 22
D. 26
E. 34
07 Dec 2020, 08:11
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Given: A group of friends on a hoke split up into smaller groups. When they formed groups of three, one person was left out. When they formed groups of four, two people were left out. When they formed groups of six, four people were left out.
Asked: If the group contains at least a dozen people, what is the smallest possible number of people in the group?
Let the number of people on the group be x
x = 3k + 1 = 4m + 2 = 6l + 4; where k,m & l are integers
x > 12
\(x_{min} = ?\)
Case 1:
3k + 1 = 4m + 2
3k - 4m = 1
(k,m) = {(3,2),(7,5),(11,8)....}
x = {10, 22, 34, ....}
Case 2:
4m + 2 = 6l + 4
4m - 6l = 2
2m - 3l = 1
(l,m) = {(2,1),(5,3),(8,5),...}
x = {6,14,22,.....}
x = 22 is common to both the cases and is greater than 12
x = 22 = 3*7 + 1 = 4*5 + 2 = 6*3 + 4 > 12
\(x_{min} = 22\)
IMO C
Asked: If the group contains at least a dozen people, what is the smallest possible number of people in the group?
Let the number of people on the group be x
x = 3k + 1 = 4m + 2 = 6l + 4; where k,m & l are integers
x > 12
\(x_{min} = ?\)
Case 1:
3k + 1 = 4m + 2
3k - 4m = 1
(k,m) = {(3,2),(7,5),(11,8)....}
x = {10, 22, 34, ....}
Case 2:
4m + 2 = 6l + 4
4m - 6l = 2
2m - 3l = 1
(l,m) = {(2,1),(5,3),(8,5),...}
x = {6,14,22,.....}
x = 22 is common to both the cases and is greater than 12
x = 22 = 3*7 + 1 = 4*5 + 2 = 6*3 + 4 > 12
\(x_{min} = 22\)
IMO C
ananya3
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07 Dec 2020, 23:12
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IMO C
total = 3x + 1
or total = 4y + 2
or total = 6z + 4
=> the increments for the 3 combinations are 3,4,6
and total > 12
starting with the highest increment
6z+4
check for z =2 since total > 12
total = 16
but 16-2 is not divisible by 4
next z = 3
6z+4 = 22
4*5 + 2 = 22
and 3*7+1 = 22
=> smallest possible number of people = 22
total = 3x + 1
or total = 4y + 2
or total = 6z + 4
=> the increments for the 3 combinations are 3,4,6
and total > 12
starting with the highest increment
6z+4
check for z =2 since total > 12
total = 16
but 16-2 is not divisible by 4
next z = 3
6z+4 = 22
4*5 + 2 = 22
and 3*7+1 = 22
=> smallest possible number of people = 22
