1990 AHSME Problems/Problem 19
Problem
For how many integers between and is the improper fraction in lowest terms?
Solution
What we want to know is for how many is We start by setting for some arbitrary . This shows that evenly divides . Next we want to see under which conditions also divides . We know from the previous statement that and thus Next we simply add to get However, we also want which leads to from the previous statement. Since from that statement divides evenly, must be of the form , for some arbitrary integer . After this, we can set and Finally, we must find the largest such that This is a simple linear equality for which the answer is , or . Solution by hjklhjklhjkl2008.
See also
1990 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 18 |
Followed by Problem 20 | |
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