Maths Olympiad Prep

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Number theory Difficulty 1.9 Junior Find the answer

On June 1, a group of students is standing in rows, with 15 students in each row. On June 2, the same group is standing with all of the students in one long row. On June 3, the same group is standing with just one student in each row. On June 4, the same group is standing with 6 students in each row. This process continues through June 12 with a different number of students per row each day. However, on June 13, they cannot find a new way of organizing the students. What is the smallest possible number of students in the group?

Pick one

Solution

The text suggests that the number of students in the group has 1212 factors, since each arrangement is a factor. The smallest integer with 1212 factors is 2235=(C) 602^2\cdot3\cdot5=\boxed{\textbf{(C) }60}.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.