2006 Alabama ARML TST Problems/Problem 12

Problem

Yoda begins writing the positive integers starting from 1 and continuing consecutively as he writes. When he stops, he realizes that there is no set of 5 composite integers among the ones he wrote such that each pair of those 5 is relatively prime. What’s the largest possible number Yoda could have stopped on?

Solution

The least group of 5 positive consecutive integers with the property that each pair of those is relatively prime is the set $\{ 2^2, 3^3, 5^2, 7^2, 11^2\}$. Therefore, the largest number he could have stopped on is $11^2-1=\boxed{120}$.

See also

2006 Alabama ARML TST (Problems)
Preceded by:
Problem 11
Followed by:
Problem 13
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