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A teacher organized a field trip for a group of children, with the total number of children between 58 and 62, inclusive. If each child must be assigned to exactly one group, can the teacher divide all the children equally into groups?
(1) The number of groups is greater than 6 but fewer than 9.
(2) The total number of children is not divisible by 5.
(1) The number of groups is greater than 6 but fewer than 9.
(2) The total number of children is not divisible by 5.
01 May 2025, 04:56
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Official Solution:
A teacher organized a field trip for a group of children, with the total number of children between 58 and 62, inclusive. If each child must be assigned to exactly one group, can the teacher divide all the children equally into groups?
We are asked whether all the children can be divided equally into groups, which means (the number of children)/(the number of groups) is an integer.
(1) The number of groups is greater than 6 but fewer than 9.
So the number of groups is either 7 or 8. From the stem, the number of children is 58, 59, 60, 61, or 62. None of these is divisible by either 7 or 8. So it does not matter how many children there are. In every case, equal division into either 7 or 8 groups is not possible. Sufficient.
(2) The total number of children is not divisible by 5.
This removes 60 from consideration, leaving 58, 59, 61, and 62. If the number of children is 58 and there are 2 groups, the answer is Yes. But if the number of groups is 4, the answer is No. Not sufficient.
Answer: A
A teacher organized a field trip for a group of children, with the total number of children between 58 and 62, inclusive. If each child must be assigned to exactly one group, can the teacher divide all the children equally into groups?
We are asked whether all the children can be divided equally into groups, which means (the number of children)/(the number of groups) is an integer.
(1) The number of groups is greater than 6 but fewer than 9.
So the number of groups is either 7 or 8. From the stem, the number of children is 58, 59, 60, 61, or 62. None of these is divisible by either 7 or 8. So it does not matter how many children there are. In every case, equal division into either 7 or 8 groups is not possible. Sufficient.
(2) The total number of children is not divisible by 5.
This removes 60 from consideration, leaving 58, 59, 61, and 62. If the number of children is 58 and there are 2 groups, the answer is Yes. But if the number of groups is 4, the answer is No. Not sufficient.
Answer: A
